Siste publikasjoner - Risiko og Stokastikk

• Imkeller, Peter; Pamen, Olivier Menoukeu & Proske, Frank Norbert (2026). TAKAGI TYPE FUNCTIONS AND DYNAMICAL SYSTEMS: THE SMOOTHNESS OF THE SBR MEASURE AND THE EXISTENCE AND SMOOTHNESS OF LOCAL TIME. arXiv.org. ISSN 2331-8422. doi: https:/arxiv.org/pdf/2107.07185v4. Fulltekst i vitenarkiv Vis sammendrag

  We investigate the occupation measures and local times of Takagi-type functions with roughness parameter γ, which are Hölder continuous with exponent H = log γ / log(1/2). Analytical insight is obtained by embedding these functions into a dynamical system related to the baker transform, whose global attractor is the graph of the Takagi function. The associated stable manifolds support Sinai-Bowen-Ruelle (SBR) measures, which we identify with the laws of certain symmetric Bernoulli convolutions. Dually, where duality is induced by time reversal, we derive a representation of the Takagi-type curves centred around the stable fibres in terms of Bernoulli convolutions, thereby relating SBR measures to occupation measures. While Bernoulli convolutions belong to the first Rademacher chaos, we show that the occupation measure is naturally represented as a functional in the second Rademacher chaos within the framework of non-Gaussian Malliavin calculus. Using a Fourier-analytic criterion together with variants of Weyl’s equidistribution theorem, we prove that Takagi-type curves admit square-integrable local times for γ = 2^(-1/m), m ≥ 7, and that the same conclusion holds for drifted Takagi curves for almost every γ ∈ (1/2, 1).

• Mirafzali, Ehsan; Proske, Frank Norbert & Marinescu, Razvan (2026). Logarithmic derivatives of variational and singular stochastic partial differential equations. arXiv.org. ISSN 2331-8422. arXiv:2608.21805 (math). doi: https:/arxiv.org/pdf/2608.21805. Fulltekst i vitenarkiv Vis sammendrag

  For a stochastic partial differential equation posed on a Gelfand triple and satisfying the fully local monotone conditions of Röckner, Shang and Zhang, we compute the logarithmic derivative of the law of the solution at a fixed time along a prescribed direction of the state space. The formula is intrinsic, being expressed through the Hilbert-Schmidt Malliavin derivative Φr=DrX(t) and the covariance γt=∫t0ΦrΦ∗rdr alone, so that neither the inversion of the first variation used in finite dimensions nor uniform Malliavin-Sobolev bounds on the Tikhonov family are called upon. It is obtained from an integration-by-parts identity on an abstract Hilbert space, a Moore-Penrose construction of a covering field on Wiener space, and a trace formula for the Tikhonov limit, specialised to the equation through the representation Φr=Y(t,r)B(r,X(r)) of the Malliavin derivative by the first variation; the stochastic p-Laplacian and the two-dimensional Navier-Stokes equation are treated in detail. Beyond the variational class, a scalar reduction gives an integration-by-parts identity for the law of a pairing ⟨u(t),φ⟩ which passes to the renormalised limit for the singular equations of Bruned, Chandra, Chevyrev and Hairer, and which is represented by a logarithmic derivative under second-order Malliavin smoothness and negative-moment hypotheses.

• Agram, Nacira; Øksendal, Bernt Karsten; Proske, Frank Norbert & Tymochenko, Olena (2026). Optimal Stopping for Systems Driven by the Brownian Sheet. arXiv.org. ISSN 2331-8422. arXiv:2603.12853 (math). doi: https:/arxiv.org/abs/2603.12853. Fulltekst i vitenarkiv Vis sammendrag

  We investigate optimal stopping problems for systems driven by the Brownian sheet. Our analysis is divided into two parts. In the first part, we derive explicit solutions to two optimal stopping problems for the exponentially discounted Brownian sheet. The first problem consists in determining the optimal two parameter first hitting point τ = (τ1,τ2) that maximizes E e−ρτ1τ2 h B(τ1,τ2), where ρ > 0 is a discount factor and h is a reward function. Restricting attention to first hitting points of levels, we obtain a closed form characterization of the optimal stopping threshold. In particular, for linear rewards h(y) = y, the optimal level is y= (2ρ)−1/2 . The second problem concerns optimal stopping of the integrated discounted Brownian sheet: E τ1 0 τ2 0 e−ρtxB(t,x) dtdx . We show that the optimal first hitting level is strictly positive, and provide an explicit representation of the value function in terms of the exponential integral function. The optimal threshold is characterized as the unique solution of a nonlinear equation derived from a Laplace transform identity for the product τ1τ2.The result is surprising, in view of the (well-known) fact that the solution of the corresponding one-parameter problem is to stop at the first time t when B(t) hits the negative level y=−(2ρ)−1/2 . In the second and main part of the paper, we develop a general potential theoretic framework for two parameter optimal stopping problems associated with stochastic partial differential equations (SPDEs) driven by the Brownian sheet. For a class of Itˆo sheets and time space homogeneous SPDEs, we prove that the value function is the least superharmonic majorant of the reward and that the optimal stopping point is given by the first exit point from the continuation region. This yields a general existence theorem for optimal stopping points in the plane, extending classical one parameter theory to the two parameter Brownian sheet setting.

• Mirafzali, Ehsan; Proske, Frank Norbert; Venturi, Daniele & Marinescu, Razvan (2026). SCORE-BASED DIFFUSION MODELS IN INFINITE DIMENSIONS: A MALLIAVIN CALCULUS PERSPECTIVE. Journal of Machine Learning for Modeling and Computing. ISSN 2689-3967. 7(2). doi: https:/www.dl.begellhouse.com/download/article/7de76a605645f298/JMLMC0702(2)-63274.pdf. Fulltekst i vitenarkiv Vis sammendrag

  We study score-based diffusion modelling in infinite-dimensional separable Hilbert spaces through Malliavin calculus, extending the analysis of generative models beyond the finite-dimensional set- ting. The forward diffusion process is formulated as a linear stochastic partial differential equa- tion (SPDE) driven by space–time coloured noise with a trace-class covariance operator, ensuring well-posedness in arbitrary spatial dimensions. Building on Malliavin calculus and an infinite- dimensional extension of the Bismut–Elworthy–Li formula, we derive a closed-form expression for the logarithmic derivative of the transition measure along Cameron–Martin directions, which serves as the natural infinite-dimensional analogue of the score function. Our operator-theoretic approach preserves the intrinsic geometry of Hilbert spaces and accommodates general trace-class operators, thereby incorporating spatially correlated noise without assuming semigroup invertibility. We vali- date the derived score formula numerically for several classes of linear SPDEs in both one and two spatial dimensions using spectral methods.

• Compagnoni, Enea Monzio; Islamov, Rustem; Proske, Frank Norbert; Lucchi, Aurelien; Orvieto, Antonio & Gorbunov, Eduard (2026). On the Interaction of Batch Noise, Adaptivity, and Compression, under (L0, L1)-Smoothness: An SDE Approach. Proceedings of Machine Learning Research (PMLR). ICML 2026. doi: https:/openreview.net/attachment?id=Pmsc4yytlf&name=pdf. Fulltekst i vitenarkiv Vis sammendrag

  Distributed stochastic optimization intertwines (i) stochastic gradient noise, (ii) communication compression, and (iii) adaptive/normalized updates. While each factor has been studied in isolation, their joint effect under realistic assumptions remains poorly understood. In this work, we develop a unified theoretical framework for Distributed Compressed SGD (DCSGD) and its sign variant Distributed SignSGD (DSignSGD) under the recently introduced (L0,L1)- smoothness condition. From a conceptual perspective, we show that the first- and second-order modified equations from the literature do not accurately model the discrete-time stepsize/stability restrictions, especially under (L0,L1)-smoothness. From a technical perspective, we propose new first-order SDEs by carefully incorporating curvature-dependent terms into their drift: This helps capture the fine-grained relationship between learning rate restrictions, gradient noise, compression, and the geometry of the loss landscape. Importantly, we do so under general gradient noise assumptions, including heavy-tailed and affine-variance regimes, which extend beyond the classical bounded-variance setting. Our results suggest that normalizing the updates of DCSGD emerges as a natural condition for stability, with the degree of normalization precisely determined by the gradient noise structure, the landscape’s regularity, and the compression rate. In contrast, DSignSGD converges even under heavy-tailed noise with standard learning rate schedules. Together, these findings offer both new theoretical insights and perspectives, and practical guidance.

• Morale, Daniela; Tarquini, Leonardo & Ugolini, Stefania (2026). A Feynman--Kac representation of a non-conservative and path-dependent nonlinear reaction-diffusion-advection system. arXiv. doi: 10.48550/ARXIV.2407.19301. Fulltekst i vitenarkiv Vis sammendrag

  We provide a probabilistic interpretation of a weakly parabolic PDE--ODE system with a reaction term, which makes the dynamics non-conservative. As a consequence, the solution is represented as the density of a sub-probability measure solving a Feynman--Kac-type equation, where the time-marginal law of the underlying process is weighted by a survival probability induced by the reaction. This leads to a coupled stochastic formulation consisting of a non-Markovian stochastic differential equation with path-dependent coefficients and the associated Feynman--Kac-type equation. We prove well-posedness of the resulting stochastic system. Finally, we introduce the corresponding interacting particle system and show that its empirical measure, suitably weighted by the survival probability associated with the reaction rate, converges to the limiting sub-probability.

• Morale, Daniela; Tarquini, Leonardo & Ugolini, Stefania (2026). Path-dependent McKean PDEs with reaction: a discussion on probabilistic interpretations and particle approximations. arXiv. doi: 10.48550/ARXIV.2602.07255. Fulltekst i vitenarkiv Vis sammendrag

  We discuss and compare two probabilistic approaches for associating a stochastic differential equation with a McKean-type partial differential equation featuring a reaction term and path-dependent coefficients. The non-conservative nature of the macroscopic dynamics leads to two possible interpretations of the sub-probability measure and of the associated SDE equation at the microscale: on the one hand, as a measure-valued solution of a Feynman-Kac-type equation; on the other hand, as the sub-probability associated with an SDE defined up to a survival time with a reaction-dependent rate. These different interpretations give rise to two different microscopic stochastic models and therefore to two different techniques of probabilistic analysis. Finally, by considering the interacting particle systems associated with both models, we discuss how their empirical densities provide two different kernel estimators for the PDE solution.

• Benth, Fred Espen; Ortiz-Latorre, Salvador & Tarquini, Leonardo (2026). Signature McKean-Vlasov stochastic differential equations. arXiv. doi: 10.48550/ARXIV.2606.22964. Fulltekst i vitenarkiv Vis sammendrag

  McKean-Vlasov-type stochastic differential equations (SDEs) are characterized by coefficients depending on both the state and the law of the solution. In this work, we focus on a class of such equations where the coefficients depend on a linear combination of the expected signature of the geometric p-rough path lift of its solution, with $p\in(2,3)$. After establishing the strong existence and uniqueness of a solution, we prove how such an equation can approximate a general class of path-dependent McKean-Vlasov SDEs. Finally, we consider the associated particle system and propagation of chaos is established.

• Lozano, Pere Diaz & Nunno, Giulia Di (2026). Hölder regularity for backward stochastic Volterra integral equations and applications to numerical schemes. arXiv.org. ISSN 2331-8422. doi: 10.48550/ARXIV.2607.04012. Fulltekst i vitenarkiv Vis sammendrag

  We prove a Hölder-type regularity estimate for the martingale integrand of a backward stochastic Volterra integral equation (BSVIE). The estimate is formulated in Lp(Ω) after averaging in L2 over the first time variable, and gives an averaged Hölder estimate of order 1/2 in the second time variable. Our approach is based on the approximation of the BSVIE by a system of BSDEs. For this system, we establish a uniform regularity estimate for the martingale component using Malliavin calculus, and then pass to the limit to obtain the result for the BSVIE. We allow for a general Malliavin differentiable free term and generator. In particular, neither is assumed to come from a forward stochastic differential equation or to have a specific functional form. We also propose an explicit Euler scheme for the approximating BSDE system and show that the regularity estimate yields a convergence rate for the resulting discrete approximation of the BSVIE.

• Lozano, Pere Diaz & Kloster, Thomas K. (2026). A Wiener Chaos Approach to Martingale Modelling and Implied Volatility Calibration. arXiv.org. ISSN 2331-8422. doi: 10.48550/ARXIV.2602.16232. Fulltekst i vitenarkiv Vis sammendrag

  Calibration to a surface of option prices requires specifying a suitably flexible martingale model for the discounted asset price under a risk-neutral measure. Assuming Brownian noise and mean-square integrability, we construct an over-parameterized model based on the martingale representation theorem. In particular, we approximate the terminal value of the martingale via a truncated Wiener--chaos expansion and recover the intermediate dynamics by computing the corresponding conditional expectations. Using the Hermite-polynomial formulation of the Wiener chaos, we obtain easily implementable expressions that enable fast calibration to a target implied-volatility surface. We illustrate the flexibility and expressive power of the resulting model through numerical experiments on both simulated and real market data.

• Diaz-Lozano, Pere & Nunno, Giulia Di (2026). Deep operator BSDE: A numerical scheme to approximate solution operators. The Annals of Applied Probability. ISSN 1050-5164. 36(3), s. 2594–2625. doi: 10.1214/25-aap2289. Fulltekst i vitenarkiv Vis sammendrag

  Motivated by dynamic risk measures and conditional g-expectations, in this work we propose a numerical method to approximate the solution operator given by a backward stochastic differential equation (BSDE). The main ingredients for this are the Wiener chaos decomposition and the classical Euler scheme for BSDEs. We show convergence of this scheme under very mild assumptions, and provide a rate of convergence in more restrictive cases. We then implement it using neural networks, and we present several numerical examples where we can check the accuracy of the method.

• Morale, Daniela; Tarquini, Leonardo & Ugolini, Stefania (2026). Killed path-dependent McKean–Vlasov SDEs for a probabilistic representation of non-conservative Mckean PDEs. Stochastics and Dynamics. ISSN 0219-4937. doi: 10.1142/s0219493726500218. Fulltekst i vitenarkiv Vis sammendrag

  A McKean–Vlasov stochastic differential equation subject to killing associated to a regularised non-conservative and path-dependent nonlinear parabolic partial differential equation is studied. The existence and pathwise uniqueness of a strong solution and the regularity properties of its sub-probability law are proved. The density of such a law may be seen as a weak solution of the considered PDE. The well-posedness of the associated particle system is also discussed.

• Benth, Fred Espen; Harang, Fabian Andsem & Straum, Fride Josefine Emberland (2026). Universal Approximation on Non-geometric Rough Paths and Applications to Financial Derivatives Pricing. Applied Mathematical Finance. ISSN 1350-486X. doi: 10.1080/1350486X.2026.2659017. Fulltekst i vitenarkiv Vis sammendrag

  We present a novel perspective on the universal approximation theorem for rough path functionals, introducing a polynomial-based approximation class. We extend universal approximation to non-geometric rough paths within the tensor algebra. This development addresses critical needs in finance, where no-arbitrage conditions necessitate Itô integration. Furthermore, our findings motivate a hypothesis for payoff functionals in financial markets, allowing straightforward analysis of signature payoffs proposed in Arribas (Derivatives Pricing Using Signature Payoffs; 2018).

• Banos, David R. & Salvesen, Kristopher (2026). A critical statistical analysis of Norwegian transgender referral data. Journal of Gender Studies. ISSN 0958-9236. doi: 10.1080/09589236.2026.2671868. Fulltekst i vitenarkiv Vis sammendrag

  This article critically analyzes referral data for gender-affirming care in Norway, focusing on two publicly available datasets: referrals to the National Center for Gender Incongruence (NCGI) and registrations at the Municipal Health Clinic for Gender and Sexuality (HCGS) in Oslo. Referral trends to gender identity clinics are often framed as sudden, sharp, or alarming increases, yet such interpretations are rarely assessed against demographic baselines or sociopolitical context. We examine whether Norwegian referral patterns support claims of exponential growth, whether structural changes can be detected statistically, and how observed counts compare with plausible population-level expectations. Using descriptive trend analysis and change point detection, we identify structural shifts around 2016 and 2020, corresponding to the Norwegian Gender Recognition Act and the COVID-19 pandemic. We find no evidence of sustained exponential growth. Instead, the data indicate gradual increases, periods of stabilization, and context-dependent interruptions. Monte Carlo simulations for the NCGI child and adolescent sample suggest that observed referrals remain within, or below, ranges implied by comparable population estimates. The article offers a transferable framework for contextualizing referral data and argues for careful, non-pathologizing interpretation of trans health statistics.

• Heupink, Lieke Fleur; Nakstad, Espen Rostrup; Lurås, Hilde; Wiik, Pål; Lippestad, Kristine & Benth, Fred Espen [Vis alle 7 forfattere av denne artikkelen] (2026). Short term forecast of new daily pandemic hospitalizations: A time series model for a single hospital. Epidemics. ISSN 1755-4365. 54. doi: 10.1016/j.epidem.2026.100894. Fulltekst i vitenarkiv
• Jolis, Maria; Ortiz-Latorre, Salvador & Quer-Sardanyons, Lluís (2026). Convergence in law for quasi-linear SPDEs. Stochastics and Partial Differential Equations: Analysis and Computations. ISSN 2194-0401. doi: 10.1007/s40072-026-00419-x. Fulltekst i vitenarkiv Vis sammendrag

  We consider the quasi-linear stochastic wave and heat equations in $\mathbb{R}^d$ with $d\in \{1,2,3\}$ and $d\geq 1$, respectively, and perturbed by an additive Gaussian noise which is white in time and has a homogeneous spatial correlation with spectral measure $\mu_n$. We allow the Fourier transform of $\mu_n$ to be a genuine distribution. Let $u^n$ be the mild solution to these equations. We provide sufficient conditions on the measures $\mu_n$ and the initial data to ensure that $u^n$ converges in law, in the space of continuous functions, to the solution of our equations driven by a noise with spectral measure $\mu$, where $\mu_n\to\mu$ in some sense. We apply our main result to various types of noises, such as the anisotropic fractional noise. We also show that we cover existing results in the literature, such as the case of Riesz kernels and the fractional noise with $d=1$.

• Agram, Nacira; Øksendal, Bernt; Proske, Frank Norbert & Tymoshenko, Olena (2026). SPDE games driven by a Brownian sheet with applications to pollution minimization. Journal of Dynamics and Games. ISSN 2164-6066. 0(0), s. 0–0. doi: 10.3934/jdg.2026009. Fulltekst i vitenarkiv Vis sammendrag

  This paper investigates a nonzero-sum stochastic differential game with players in the context of shared spatial-domain pollution control. The pollution dynamics are modeled by a stochastic partial differential equation (SPDE) driven by a Brownian sheet, capturing spatial and temporal environmental randomness. Each player, representing a distinct region or decision-maker, minimizes an individual cost functional balancing pollution penalties against control implementation costs. Using a stochastic maximum principle in the space-time setting, we derive necessary and sufficient conditions for Nash equilibrium strategies. A novel family of adjoint variables, , is introduced; these variables have no counterpart in formulations driven by the classical one-parameter Brownian motion. Two illustrative examples demonstrate how spatial interactions and stochastic perturbations shape the equilibrium strategies and collective environmental outcomes.

• Banos, David Ruiz; Ortiz-Latorre, Salvador & Font, Oriol Zamora (2026). A functional variational approach to pricing path dependent insurance policies. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. Fulltekst i vitenarkiv Vis sammendrag

  The main purpose of this work is the derivation of a path-dependent partial differential equation for the calculations of equity-linked insurance policies, where the payment stream may depend on the whole past history of the financial asset. To this end, we employ variational techniques from the theory of functional Itô calculus.

• Coffie, Emmanuel; Menoukeu-Pamen, Olivier & Proske, Frank Norbert (2026). On the analysis of a singular stochastic Volterra differential equation driven by a Wiener noise. Electronic Communications in Probability. 31, s. 1–13. doi: 10.1214/26-ECP782. Fulltekst i vitenarkiv Vis sammendrag

  In this article, we construct unique strong solutions to a class of stochastic Volterra differential equations driven by a singular drift vector field and a Wiener noise. Further, we examine the Sobolev differentiability of the strong solution with respect to its initial value.

• Imkeller, Peter; Pamen, Olivier Menoukeu & Proske, Frank Norbert (2025). Geometric properties of rough curves via dynamical systems: SBR measure, local time, Rademacher chaos and number theory Peter Imkeller. Oberwolfach Reports. ISSN 1660-8933. 53. doi: https:/publications.mfo.de/bitstream/handle/mfo/4402/OWR_2025_53.pdf?sequence=1&isAllowed=y. Fulltekst i vitenarkiv
• Blanco, Sara Anna Solanilla (2025). Local sensitivity analysis of heating degree day and cooling degree day temperature derivative prices. Quantitative finance (Print). ISSN 1469-7688. 25(4), s. 653–670. doi: 10.1080/14697688.2025.2471347. Fulltekst i vitenarkiv Vis sammendrag

  We study the local sensitivity of heating degree day (HDD) and cooling degree day (CDD) temperature futures and option prices with respect to perturbations in the deseasonalized temperature or in one of its derivatives up to a certain order determined by the continuous-time autoregressive process modelling the deseasonalized temperature in the HDD and CDD indexes. We also consider an empirical case where a continuous-time process of autoregressive order 3 is fitted to New York temperatures and we perform a study of the local sensitivity of these financial contracts and a posterior analysis of the results.

• Rahman, Ghaus ur; Tymoshenko, Olena & Nunno, Giulia Di (2025). Insights on Stochastic Dynamics for Transmission of Monkeypox: Biological and Probabilistic Behavior. Mathematical Methods in the Applied Sciences. ISSN 0170-4214. 48(15), s. 14316–14333. doi: 10.1002/mma.11180. Fulltekst i vitenarkiv
• Panichek, O. V. & Tymoshenko, Olena (2025). LONG-TERM DYNAMICS OF A STOCHASTIC SIRV MODEL FOR TB WITH RANDOM DISTURBANCES. Journal of Optimization, Differential Equations and their Applications. ISSN 2312-4547. 33(2), s. 17–34. doi: 10.15421/142512. Fulltekst i vitenarkiv
• Benth, Fred Espen; Deelstra, Griselda & Kozpınar, Sinem (2025). Pricing energy quanto options: a regime-switching framework with stochastic interest rates. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. doi: 10.1080/17442508.2025.2541874. Fulltekst i vitenarkiv Vis sammendrag

  Energy quanto options are risk management instruments gaining a lot of attraction from the actors in energy markets due to their nature, which allows to handle volumetric and price risk simultaneously, in contrast to typical contracts that solely take price risk into consideration. The payoff of these options is constructed in such a way that it resembles the product of the payouts from two options, each with an energy-related underlying. If the underlying assets are considered to be energy futures contracts, which are more easily traded assets than temperature and energy spots themselves, the payoff of the energy quanto option can be treated as the product of the payoff of two European vanilla options. This paper investigates the valuation of energy quanto options written on temperature and electricity futures in the presence of regime-switching stochastic interest rates. The electricity futures price is governed by a continuous Markov-modulated model with mean reversion, and the dynamics of temperature futures evolve without regime-switching but retain seasonality and mean reversion. Using a change of measure and fast Fourier transform techniques, we obtain a pricing formula for energy quanto options on futures. Numerical examples are carried out to demonstrate the implementation of the model.

• Agram, Nacira; Benth, Fred Espen & Pucci, Giulia (2025). Installation of renewable capacities to meet energy demand and emission constraints under uncertainty. IMA Journal of Management Mathematics. ISSN 1471-678X. 37(1), s. 39–60. doi: 10.1093/imaman/dpaf023. Fulltekst i vitenarkiv Vis sammendrag

  This paper focuses on minimizing the costs of installing renewable energy capacity while meeting emission constraints under uncertainty in both energy demand and renewable production. We consider a setting where decision-makers must determine when and how much renewable capacity to install, balancing investment costs with future emissions. Our optimization problem combines cost minimization with a probabilistic constraint on total accumulated emissions, reflecting regulatory limits that may be exceeded only with small probability. We examine different investment strategies, allowing for one or multiple installation times, and provide explicit solutions in simplified cases. Our main insight is that, under reasonable assumptions on costs and uncertainty, a single, well-timed investment is optimal and may be delayed to reduce costs when uncertainty and discounting are accounted for. These results challenge common stepwise installation strategies and suggest that committing to a single large investment, possibly postponed, may be more cost-effective and efficient in reaching emission targets. Our findings offer practical guidance for policymakers and energy planners on how to balance costs, timing and environmental goals when expanding renewable energy capacity under uncertainty.

• Lozano, Pere Diaz & Nunno, Giulia Di (2025). An Euler scheme for BSDEs via the Wiener chaos decomposition. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv Vis sammendrag

  The Euler scheme is a standard time discretization for BSDEs, but its implementation hinges on approximating conditional expectations and the associated martingale terms at each time step. We propose an implementation based on the Wiener chaos decomposition to approximate these quantities. In contrast to many numerical schemes that rely on a forward-backward (Markovian) structure, our approach accommodates arbitrary $\mathcal{F}_T$-measurable square-integrable terminal conditions. We provide a comprehensive convergence analysis and illustrate the method on several numerical examples.

• Ehsan, Mirafzali,; Utkarsh, Gupta,; Patrick, Wyrod,; Proske, Frank Norbert; Daniele, Venturi, & Razvan, Marinescu, (2025). [2503.16917] Malliavin Calculus for Score-based Diffusion Models. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv Vis sammendrag

  We introduce a new framework based on Malliavin calculus to derive exact analytical expressions for the score function $\nabla \log p_t(x)$, i.e., the gradient of the log-density associated with the solution to stochastic differential equations (SDEs). Our approach combines classical integration-by-parts techniques with modern stochastic analysis tools, such as Bismut's formula and Malliavin calculus, and it works for both linear and nonlinear SDEs. In doing so, we establish a rigorous connection between the Malliavin derivative, its adjoint, the Malliavin divergence (Skorokhod integral), and diffusion generative models, thereby providing a systematic method for computing $\nabla \log p_t(x)$. In the linear case, we present a detailed analysis showing that our formula coincides with the analytical score function derived from the solution of the Fokker--Planck equation. For nonlinear SDEs with state-independent diffusion coefficients, we derive a closed-form expression for $\nabla \log p_t(x)$. We evaluate the proposed framework across multiple generative tasks and find that its performance is comparable to state-of-the-art methods. These results can be generalised to broader classes of SDEs, paving the way for new score-based diffusion generative models.

• Ehsan, Mirafzali,; Proske, Frank Norbert; Utkarsh, Gupta,; Daniele, Venturi, & Razvan, Marinescu, (2025). [2507.05550] A Malliavin calculus approach to score functions in diffusion generative models. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv Vis sammendrag

  Score-based diffusion generative models have recently emerged as a powerful tool for modelling complex data distributions. These models aim at learning the score function, which defines a map from a known probability distribution to the target data distribution via deterministic or stochastic differential equations (SDEs). The score function is typically estimated from data using a variety of approximation techniques, such as denoising or sliced score matching, Hyvärien's method, or Schrödinger bridges. In this paper, we derive an exact, closed-form, expression for the score function for a broad class of nonlinear diffusion generative models. Our approach combines modern stochastic analysis tools such as Malliavin derivatives and their adjoint operators (Skorokhod integrals or Malliavin Divergence) with a new Bismut-type formula. The resulting expression for the score function can be written entirely in terms of the first and second variation processes, with all Malliavin derivatives systematically eliminated, thereby enhancing its practical applicability. The theoretical framework presented in this work offers a principled foundation for advancing score estimation methods in generative modelling, enabling the design of new sampling algorithms for complex probability distributions. Our results can be extended to broader classes of stochastic differential equations, opening new directions for the development of score-based diffusion generative models.

• Ehsan, Mirafzali,; Proske, Frank Norbert; Daniele, Venturi, & Razvan, Marinescu, (2025). [2508.20316] Score-Based Diffusion Models in Infinite Dimensions: A Malliavin Calculus Perspective. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv Vis sammendrag

  We develop a new mathematical framework for score-based diffusion modelling in infinite-dimensional separable Hilbert spaces, through Malliavin calculus, extending generative modelling techniques beyond the finite-dimensional setting. The forward diffusion process is formulated as a linear stochastic partial differential equation (SPDE) driven by space--time coloured noise with a trace-class covariance operator, ensuring well--posedness in arbitrary spatial dimensions. Building on Malliavin calculus and an infinite-dimensional extension of the Bismut--Elworthy--Li formula, we derive an exact, closed-form expression for the score function (the Fréchet derivative of the log-density) without relying on finite-dimensional projections or discretisation. Our operator--theoretic approach preserves the intrinsic geometry of Hilbert spaces and accommodates general trace-class operators, thereby incorporating spatially correlated noise without assuming semigroup invertibility. The framework also connects score estimation to functional data analysis, enabling computational strategies such as kernel methods and neural operator architectures.

• Compagnoni, Enea Monzio; Islamov, Rustem; Proske, Frank Norbert & Lucchi, Aurelien (2025). Unbiased and Sign Compression in Distributed Learning: Comparing Noise Resilience via SDEs. Proceedings of Machine Learning Research (PMLR). 258. doi: https:/proceedings.mlr.press/v258/compagnoni25a.html. Fulltekst i vitenarkiv Vis sammendrag

  Distributed methods are essential for handling machine learning pipelines comprising large-scale models and datasets. However, their benefits often come at the cost of increased communication overhead between the central server and agents, which can become the main bottleneck, making training costly or even unfeasible in such systems. Compression methods such as quantization and sparsification can alleviate this issue. Still, their robustness to large and heavy-tailed gradient noise, a phenomenon sometimes observed in language modeling, remains poorly understood. This work addresses this gap by analyzing Distributed Compressed SGD (DCSGD) and Distributed SignSGD (DSignSGD) using stochastic differential equations (SDEs). Our results show that DCSGD with unbiased compression is more vulnerable to noise in stochastic gradients, while DSignSGD remains robust, even under large and heavy-tailed noise. Additionally, we propose new scaling rules for hyperparameter tuning to mitigate performance degradation due to compression. These findings are empirically validated across multiple deep learning architectures and datasets, providing practical recommendations for distributed optimization.

• Nunno, Giulia Di & Yurchenko-Tytarenko, Anton (2025). Sandwiched Volterra volatility model: Markovian approximations and hedging. Finance and Stochastics. ISSN 0949-2984. 30, s. 277–325. doi: 10.1007/s00780-025-00584-2. Fulltekst i vitenarkiv Vis sammendrag

  We consider stochastic volatility dynamics driven by a general Hölder-continuous Volterra-type noise and with unbounded drift. For these so-called SVV models, we consider the explicit computation of quadratic hedging strategies. While the theoretical hedge is well known in terms of the non-anticipating derivative for all square-integrable claims, the fact that these models are typically non-Markovian provides a challenge in the direct computation of conditional expectations at the core of the explicit hedging strategy. To overcome this difficulty, we propose a Markovian approximation of the model which stems from an adequate approximation of the kernel in the Volterra noise. We study the approximation of the volatility, of the prices and of the optimal mean-square hedge. We provide corresponding error estimates. The work is completed with numerical simulations.

• Bogso, Antoine Marie; Pamen, Olivier Menoukeu & Proske, Frank Norbert (2025). Strong Solutions of Fractional Brownian Sheet-Driven Stochastic Differential Equations with Integrable Drift. Journal of theoretical probability. ISSN 0894-9840. 39(1). doi: 10.1007/s10959-025-01470-5. Fulltekst i vitenarkiv Vis sammendrag

  We prove the existence of a unique Malliavin differentiable strong solution to a stochastic differential equation on the plane with merely integrable coefficients and driven by the fractional Brownian sheet with Hurst parameters less than 1/2. The proof of this result relies on a compactness criterion for square-integrable Wiener functionals from Malliavin calculus (Da Prato in CR Acad Sci Paris Série 1, Mathématique 315:1287–1291, 1992), variational techniques developed in the case of fractional Brownian motion (Baños in J Dyn Diff Equat 32:1819–1866, 2020) and the concept of sectorial local nondeterminism introduced by Khoshnevisan (Khoshnevisan in Trans Amer Math Soc 359:3125–3151, 2007). The latter concept enables us to improve the bound of the Hurst parameter; compare with (Baños in J Dyn Diff Equat 32:1819–1866, 2020).

• Hvidsten, Tobias Verheugen; Roithner, Maximilian; Benth, Fred Espen & Zeyringer, Marianne (2025). Driving towards net-zero: the impact of electric vehicle flexibility participation on a future Norwegian electricity system. Environmental Research: Infrastructure and Sustainability. 5(2). doi: 10.1088/2634-4505/ade0e6. Fulltekst i vitenarkiv Vis sammendrag

  Electric vehicle (EV) batteries have a proven flexibility potential which could serve as an alternative to conventional electricity storage solutions. EV batteries could support the balancing of supply and demand, and the integration of variable renewable energy into the electricity system. The flexibility potential from EVs, in distinction to conventional battery storage, depends on the vehicle user’s willingness and opportunity to make their vehicle available for flexibility. This rate of participation is often not considered in modelling studies, despite the impact EV flexibility could have on the electricity system. This work presents a modelling study of the Norwegian electricity system using the high spatial and temporal resolution electricity system model (highRES). Two different charging strategies for EVs, flexible charging and vehicle-to-grid (V2G), are optimised in conjunction with the design and dispatch of the electricity system. By varying the rate of participation in the charging strategies we demonstrate their benefits for a future net-zero electricity system. Our findings show EVs’ potential to eliminate the need for stationary battery storage with just 50% participation in V2G. In addition, we find that the EV flexibility contributes to reductions in the total system cost by almost 4% and 15% assuming 100% participation in flexible charging and V2G, respectively.

• Roithner, Maximilian; Velasco-Herrejón, Paola; Greevenbroek, Koen van; Grochowicz, Aleksander; Vågerö, Oskar & Hvidsten, Tobias Verheugen [Vis alle 8 forfattere av denne artikkelen] (2025). Balancing act: the cost of wind restrictions in Norway’s electricity transition. Environmental Research: Energy. 2(3). doi: 10.1088/2753-3751/adde62. Fulltekst i vitenarkiv Vis sammendrag

  To meet its commitments under the Paris Agreement and reduce its dependency on energy imports, the pace, and scale of renewable energy deployment across Europe must increase dramatically over the next decade. Such a steep change in the net-zero transition will inevitably necessitate trade-offs with other societal priorities. Here we investigate a case study focused on the opposition towards onshore wind and the compromises that may need to be made to deliver its plans for deep electrification. Using an electricity system model, we explore the implications of key social and environmental dimensions shaping the future deployment of onshore wind on the costs and design of electricity systems for Norway in 2030. We find that under restrictions that allow for almost no additional onshore wind, demand cannot be met and load has to be shed. Yet, when reducing the restrictions on onshore wind or allowing for in-country transmission expansion, feasible system designs at a small fraction of that cost can be found. To meet the net-zero targets, compromises will need to be made on either wind power deployment, transmission expansion, non-electrification of industry or demand reduction.

• D'Auria, Bernardo; Nunno, Giulia Di & Salmeron, Jose A. (2025). Before and After Default: Information and Optimal Portfolio via Anticipating Calculus. Applied Mathematics and Optimization. ISSN 0095-4616. 91(2), s. 1–40. doi: 10.1007/s00245-024-10210-w. Fulltekst i vitenarkiv
• Bogso, Antoine-Marie; Dieye, Moustapha; Menoukeu-Pamen, Olivier & Proske, Frank Norbert (2025). SMOOTHNESS OF SOLUTIONS OF HYPERBOLIC STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH L∞-VECTOR FIELDS. Mathematische Annalen. ISSN 0025-5831. 393, s. 1321–1383. doi: 10.1007/s00208-025-03251-1. Fulltekst i vitenarkiv Vis sammendrag

  In this paper we are interested in a quasi-linear hyperbolic stochastic differential equation (HSPDE) when the vector field is merely bounded and measurable. Although the deterministic counterpart of such equation may be ill-posed (in the sense that uniqueness or even existence might not be valid), we show for the first time that the corresponding HSPDE has a unique (Malliavin differentiable) strong solution. Our approach for proving this result rests on: (1) tools from Malliavin calculus and (2) variational techniques introduced in Davie (International Mathematics Research Notices, vol. 2007, Duke University Press, Durham, 2007) non-trivially extended to the case of SDEs in the plane by using an algorithm for the selection of certain rectangles. As a by-product, we also obtain the Sobolev differentiability of the solution with respect to its initial value. The results derived here constitute a significant improvement of those in the current literature on SDEs on the plane and can be regarded as an analogous of the pioneering works by Zvonkin (Math. URSS Sb. 22:129–149, 1974) and Veretennikov (Theory Probab. Appl. 24:354–366, 1979) in the case of one-parameter SDEs with singular drift.

• Orlovskyi, Igor; Proske, Frank Norbert & Tymoshenko, Olena (2025). Long-time behaviors of some stochastic differential equations driven by Lévy noise. Journal of Differential Equations. ISSN 0022-0396. 442. doi: 10.1016/j.jde.2025.113561. Fulltekst i vitenarkiv Vis sammendrag

  Using key tools such as Itô's formula for general semi-martingales, moment estimates for Lévy-type stochastic integrals, and properties of regularly varying functions we find conditions under which solutions of a stochastic differential equation with jumps are almost surely asymptotically equivalent to a nonrandom function as .

• Nunno, Giulia Di; Haferkorn, Hannes; Khedher, Asma & Vanmaele, Michèle (2025). Utility maximization and change of variable formulas for time-changed dynamics. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. s. 1–26. doi: 10.1080/17442508.2025.2462815. Fulltekst i vitenarkiv
• Dehant, V.; Puica, Mihaela-Alexandra; Folgueira-López, M.; Rekier, J. & Hoolst, Hoolst Van (2025). Analytical computation of the total topographic torque at the core-mantle boundary and its impact on nutations. Geophysical Journal International. ISSN 0956-540X. 241(1), s. 474–494. doi: 10.1093/gji/ggaf050. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we examine the dynamic pressure torque acting on a bumpy core–mantle boundary (CMB) at diurnal timescale in a frame tied to the planet. This torque possibly contributes to the CMB coupling constants determined from nutation observations and could affect the interpretation of these constants in terms of different CMB coupling mechanisms. We revisit the work of Wu & Wahr (1997) who have used seismic estimates for the topography at the CMB and computed the associated pressure torque effect on nutations. These authors showed that some topography wavelengths can lead to amplifications in nutations. For example, they found that the effects on the retrograde annual nutation can be at the milliarcsecond level for a degree-5 spherical harmonics of the topography. While Wu & Wahr (1997) only go up to degree 6 in their development in spherical harmonics and use a numerical technique, we go up to degree 20 and employ an analytical approach to solve the equations and to further study the Earth’s nutations. The approach is similar to the one we used for the effects of the pressure torque on the tidal variations of the length of day (a companion paper, Puica et al. 2023). Unlike the numerical approach, this has the advantage of highlighting the mathematical dependencies between the different spherical harmonics involved in the development of the topographic torque and to highlight the frequency dependence of the results and thereby the possible resonances with inertial waves. By doing so, we can isolate and estimate the magnitude of the influence of each topographic coefficient on nutation. We show that only the core flattening may have an important role on nutation and that the other large wavelengths of the topography have a very small contribution, less than that obtained by Wu & Wahr (1997).

• Compagnoni, Enea; Liu, Tianlin; Islamov, Rustem; Proske, Frank Norbert; Orvieto, Antonio & Lucchi, Aurelien (2025). Adaptive Methods through the Lens of SDEs: Theoretical Insights on the Role of Noise. International Conference on Learning Representations (ICLR). doi: https:/proceedings.iclr.cc/paper_files/paper/2025/file/ee2e09b7a7b520bb348c0f1d5de0069d-Paper-Conference.pdf. Fulltekst i vitenarkiv
• Agram, Nacira; Øksendal, Bernt Karsten; Proske, Frank Norbert & Tymoshenko, Olena (2025). Optimal control of SPDEs driven by time-space Brownian motion. SIAM Journal of Control and Optimization. ISSN 0363-0129. 63(1), s. 546–570. doi: 10.1137/23M1595308. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we study the optimal control of systems where the state dynamics are governed by a stochastic partial differential equation (SPDE) driven by a two-parameter (time-space) Brownian motion, also referred to as a Brownian sheet. These equations can, for example, model the growth of an ecosystem under uncertainty. We first explore some fundamental properties of such linear SPDEs. Next, utilizing time-space white noise calculus, we derive both Pontryagin-type necessary and sufficient conditions for the optimality of the control. Finally, we illustrate our results by solving a linear-quadratic control problem and examining an optimal harvesting problem in the plane. Potential applications to machine learning and to managing random environmental influences are also discussed.

• Lord, Gabriel J. & Petersson, Andreas Erik (2025). PIECEWISE LINEAR INTERPOLATION OF NOISE IN FINITE ELEMENT APPROXIMATIONS OF PARABOLIC SPDEs. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 63(2), s. 542–563. doi: 10.1137/23M1574117. Fulltekst i vitenarkiv
• Agram, Nacira; Øksendal, Bernt Karsten; Proske, Frank Norbert & Tymoshenko, Olena (2025). A Kalman filter for linear systems driven by time-space Brownian sheet. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. s. 1–22. doi: 10.1080/17442508.2025.2541059. Fulltekst i vitenarkiv Vis sammendrag

  We study a linear filtering problem where the signal and observation processes are described as solutions of linear stochastic differential equations driven by time-space Brownian sheets. We derive a stochastic integral equation for the conditional value of the signal given the observation. This can be considered a time-space analogue of the classical Kalman filter. The result is illustrated with examples of the filtering problem involving noisy observations of a constant and noisy observations of the Brownian sheet.

• Nunno, Giulia Di & Giordano, Michele (2025). Maximum principles for stochastic time-changed Volterra games. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. s. 1–32. doi: 10.1080/17442508.2025.2489355. Fulltekst i vitenarkiv
• Greevenbroek, Koen van; Grochowicz, Aleksander; Zeyringer, Marianne & Benth, Fred Espen (2025). Trading off regional and overall energy system design flexibility in the net-zero transition. Nature Sustainability. doi: 10.1038/s41893-025-01556-2. Fulltekst i vitenarkiv Vis sammendrag

  Thetransitiontonet-zeroemissionsinEuropeisdeterminedbyapatchworkofcountry-levelandEUwide policy, creating coordination challenges in an interconnected system. We use an optimisation model to mapoutnear-optimal energy system designs for 2050, focussing on the planning flexibility of individual regions while maintaining overall system robustness against different weather years, cost assumptions, and land use limitations. Our results reveal extensive flexibility at a regional level, where only few technologies (solar around the Adriatic and wind on the British Isles and in Germany) cannot be substituted. National policymakers can influence renewable energy export and hydrogen strategies significantly, provided they coordinate this with the remaining European system. However, stronger commitment to solar in Southern Europe and Germany unlocks more design options for Europe overall. These results on regional trade-offs facilitate more meaningful policy discussions which are crucial in the transition to a sustainable energy system.

• Benth, Fred Espen & Eyjolfsson, Heidar (2024). Robustness of Hilbert space-valued stochastic volatility models. Finance and Stochastics. ISSN 0949-2984. 28, s. 1117–1146. doi: 10.1007/s00780-024-00542-4. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we show that Hilbert space-valued stochastic models are robust with respect to perturbations, due to measurement or approximation errors, in the underlying volatility process. Within the class of stochastic-volatility-modulated Ornstein–Uhlenbeck processes, we quantify the error induced by the volatility in terms of perturbations in the parameters of the volatility process. We moreover study the robustness of the volatility process itself with respect to finite-dimensional approximations of the driving compound Poisson process and semigroup generator, respectively, when considering operator-valued Barndorff-Nielsen and Shephard stochastic volatility models. We also give results on square root approximations. In all cases, we provide explicit bounds for the induced error in terms of the approximation of the underlying parameter. We discuss some applications to robustness of prices of options on forwards and volatility.

• Hachemi, Rahma Yasmina Moulay & Øksendal, Bernt Karsten (2024). The stochastic heat equation with fractional time and fractional time–space white noise. Expositiones mathematicae. ISSN 0723-0869. doi: 10.1016/j.exmath.2024.125625. Fulltekst i vitenarkiv
• Kalinin, Alexander; Meyer-Brandis, Thilo & Proske, Frank Norbert (2024). Stability, uniqueness and existence of solutions to McKean–Vlasov SDEs: a multidimensional Yamada–Watanabe approach. Stochastics and Dynamics. ISSN 0219-4937. 24(5). doi: 10.1142/S0219493724500394. Fulltekst i vitenarkiv Vis sammendrag

  We establish stability and pathwise uniqueness of solutions to Wiener noise driven McKean–Vlasov equations with random non-Lipschitz continuous coefficients. In the deterministic case, we also obtain the existence of unique strong solutions. By using our approach, which is based on an extension of the Yamada–Watanabe ansatz to the multidimensional setting and which does not rely on the construction of Lyapunov functions, we prove first moment and pathwise exponential stability. Furthermore, Lyapunov exponents are computed explicitly.

• Grochowicz, Aleksander; Benth, Fred Espen & Zeyringer, Marianne (2024). Spatio-temporal smoothing and dynamics of different electricity flexibility options for highly renewable energy systems—Case study for Norway. Applied Energy. ISSN 0306-2619. 356. doi: 10.1016/j.apenergy.2023.122338. Fulltekst i vitenarkiv Vis sammendrag

  In this article, we investigate the mismatch of renewable electricity production to demand and how flexibility options enabling spatial and temporal smoothing can reduce risks of variability. As a case study we pick a simplified (partial) 2-region representation of the Norwegian electricity system and focus on wind power. We represent regional electricity production and demand through two stochastic processes: the wind capacity factors are modelled as a two-dimensional Ornstein–Uhlenbeck process and electricity demand consists of realistic base load and temperature-induced load coming from a deseasonalised autoregressive process. We validate these processes, that we have trained on historical data, through Monte Carlo simulations allowing us to generate many statistically representative weather years. For the investigated realisations (weather years) we study deviations of production from demand under different wind capacities, and introduce different scenarios where flexibility options like storage and transmission are available. Our analysis shows that simulated loss values are reduced significantly by cooperation between regions and either mode of flexibility. Combining storage and transmission leads to even more synergies and helps to stabilise production levels and thus reduces likelihoods of inadequacy of renewable power systems.

• Nunno, Giulia Di; Rahman, Ghaus ur & Tymoshenko, Olena (2024). Insights on Stochastic Dynamics for Transmission of Monkeypox: Biological and Probabilistic Behaviour. arXiv.org. ISSN 2331-8422. doi: 10.48550/arXiv.2402.05594. Fulltekst i vitenarkiv
• Grid, Maroua; Agram, Nacira; Kebiri, Omar & Øksendal, Bernt Karsten (2024). Deep learning for solving initial path optimization of mean-field systems with memory. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. 97(8), s. 1016–1037. doi: 10.1080/17442508.2024.2402741. Fulltekst i vitenarkiv Vis sammendrag

  We consider the problem of finding the optimal initial investment strategy for a system modelled by a linear McKean–Vlasov (mean-field) stochastic differential equation with delay, driven by Brownian motion and a pure jump Poisson random measure. The goal is to determine the optimal initial values for the system in the period [−𝛿,0], where 𝛿>0 is a delay constant, before the system starts at t = 0. Due to the delay in the dynamics, the system will, after startup, be influenced by these initial investment values. It is known that linear stochastic delay differential equations are equivalent to stochastic Volterra integral equations. By utilizing this equivalence, we can find implicit expressions for the optimal investment. Moreover, we propose a deep neural network-based algorithm to solve the stochastic control problem with delay. Specifically, we employ a multi-layer feed-forward neural network for control modelling in the interval [−𝛿,0], and use back-propagation to train the feed-forward neural network. The gradient of the loss function is computed using stochastic gradient descent (SGD) with respect to the weights of the network.

• Vorobeva, Ekaterina; Eggen, Mari Dahl; Midtfjord, Alise Danielle; Benth, Fred Espen; Hupe, Patrick & Brissaud, Quentin [Vis alle 8 forfattere av denne artikkelen] (2024). Estimating stratospheric polar vortex strength using ambient ocean-generated infrasound and stochastics-based machine learning. Quarterly Journal of the Royal Meteorological Society. ISSN 0035-9009. 150, s. 2712–2726. doi: 10.1002/qj.4731. Fulltekst i vitenarkiv Vis sammendrag

  There are sparse opportunities for direct measurement of upper stratospheric winds, yet improving their representation in subseasonal-to-seasonal prediction models can have significant benefits. There is solid evidence from previous research that global atmospheric infrasound waves are sensitive to stratospheric dynamics. However, there is a lack of results providing a direct mapping between infrasound recordings and polar-cap upper stratospheric winds. The global International Monitoring System (IMS), which monitors compliance with the Comprehensive Nuclear-Test-Ban Treaty, includes ground-based stations that can be used to characterize the infrasound soundscape continuously. In this study, multi-station IMS infrasound data were utilized along with a machine-learning supported stochastic model, Delay-SDE-net, to demonstrate how a near-real-time estimate of the polar-cap averaged zonal wind at 1-hPa pressure level can be found from infrasound data. The infrasound was filtered to a temporal low-frequency regime dominated by microbaroms, which are ambient-noise infrasonic waves continuously radiated into the atmosphere from nonlinear interaction between counter-propagating ocean surface waves. Delay-SDE-net was trained on 5 years (2014–2018) of infrasound data from three stations and the ERA5 reanalysis 1-hPa polar-cap averaged zonal wind. Using infrasound in 2019–2020 for validation, we demonstrate a prediction of the polar-cap averaged zonal wind, with an error standard deviation of around 12 m·s compared with ERA5. These findings highlight the potential of using infrasound data for near-real-time measurements of upper stratospheric dynamics. A long-term goal is to improve high-top atmospheric model accuracy, which can have significant implications for weather and climate prediction.

• Nunno, Giulia Di & Gianin, Emanuela Rosazza (2024). Fully Dynamic Risk Measures: Horizon Risk, Time-Consistency, and Relations with BSDEs and BSVIEs. SIAM Journal on Financial Mathematics. 15(2), s. 399–435. doi: 10.1137/23M1546804. Fulltekst i vitenarkiv Vis sammendrag

  In a dynamic framework, we identify a new concept associated with the risk of assessing the financial exposure by a measure that is not adequate to the actual time horizon of the position. This will be called horizon risk. We clarify that dynamic risk measures are subject to horizon risk, so we propose to use the fully dynamic version. To quantify horizon risk, we introduce h-longevity as an indicator. We investigate these notions together with other properties of risk measures, such as normalization, restriction property, and different formulations of time-consistency. We also consider these concepts for fully dynamic risk measures generated by backward stochastic differential equations (BSDEs), backward stochastic Volterra integral equations (BSVIEs), and families of these. Within this study, we provide new results for BSVIEs, such as a converse comparison theorem and the dual representation of the associated risk measures.

• Vågerö, Oskar; Bråte, Anders; Wittemann, Alexandra Knudsen; Robinson, Jessica Yarin; Zeyringer, Marianne & Sirotko-Sibirskaya, Natalia (2024). Machine learning of public sentiments towards wind energy in Norway. Wind Energy. ISSN 1095-4244. doi: 10.1002/we.2902. Fulltekst i vitenarkiv Vis sammendrag

  Across Europe, negative public opinion has and may continue to limit the deployment of renewable energy infrastructure required for the transition to net-zero energy systems. Understanding public sentiment and its spatio-temporal variations is as such important for decision-making and for developing socially accepted energy systems. In this study, we apply a sentiment classification model based on a machine learning framework for natural language processing, NorBERT, on data collected from Twitter between 2006 and 2022 to analyse the case of wind power opposition in Norway. From the 68,828 tweets with geospatial information, we show how discussions about wind power intensified in 2018/2019 together with a trend of more negative tweets up until 2020, both on a regional level and for Norway as a whole. Furthermore, we find weak geographical clustering in our data, indicating that discussions are country wide and not dominated by specific regional events or developments. Twitter data allow for detailed insight into the temporal nature of public sentiments and extending this research to additional case studies of technologies, countries and sources of data (e.g. newspapers, other social media) may prove important to complement traditional survey research and the understanding of public sentiment.

• Nunno, Giulia Di; Mishura, Yuliya & Yurchenko-Tytarenko, Anton (2024). Option Pricing in Sandwiched Volterra Volatility Model. SIAM Journal on Financial Mathematics. 15(3), s. 824–882. doi: 10.1137/22M1521328. Fulltekst i vitenarkiv
• Sande, Åsmund Hausken (2024). Ergodic Estimation and Model Assessment for Dynamic Exceedance Times. arXiv.org. ISSN 2331-8422. doi: 10.48550/arXiv.2406.11347. Fulltekst i vitenarkiv
• Agram, Nacira; Øksendal, Bernt Karsten; Proske, Frank Norbert & Tymoshenko, Olena (2024). Fokker-Planck equations for conditional McKean-Vlasov systems driven by Brownian sheets. arXiv.org. ISSN 2331-8422. doi: 10.48550/arXiv.2412.20572. Fulltekst i vitenarkiv
• Benth, Fred Espen; Eggen, Mari Dahl & Eisenberg, Paul (2024). Ornstein–Uhlenbeck processes in Hilbert space and autoregressive moving-average time series. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. 97(1), s. 55–80. doi: 10.1080/17442508.2024.2422122. Fulltekst i vitenarkiv Vis sammendrag

  The relationship between parabolic stochastic partial differential equations and autoregressive moving average (ARMA) time series on the real line is established. This is done in light of semigroup theory, under which the parabolic stochastic partial differential equation admits an Ornstein–Uhlenbeck process in Hilbert space. Hilbert-valued AR(1) (or, ARH(1) for short) processes are shown to naturally appear from sampled Ornstein–Uhlenbeck processes. An error representation of approximating AR(1) time series for evaluated ARH(1) processes is derived for the time dimension. Further, by proper projections of ARH(1) processes into Hilbertian subspaces a spatial error representation is derived for evaluation of such projections. The result shows convergence for approximating ARMA times series with increasing spatial dimensions. A numerical example demonstrates our theoretical results for the stochastic heat equation. The results provide a functional data analysis approach to ARH(1) processes.

• Kalinin, Alexander; Meyer-Brandis, Thilo & Proske, Frank Norbert (2024). Stability, uniqueness and existence of solutions to McKean-Vlasov SDEs in arbitrary moments. Journal of theoretical probability. ISSN 0894-9840. 37, s. 2941–2989. doi: 10.1007/s10959-024-01344-2. Fulltekst i vitenarkiv Vis sammendrag

  We deduce stability and pathwise uniqueness for a McKean–Vlasov equation with random coefficients and a multidimensional Brownian motion as driver. Our analysis focuses on a non-Lipschitz continuous drift and includes moment estimates for random Itô processes that are of independent interest. For deterministic coefficients, we provide unique strong solutions even if the drift fails to be of affine growth. The theory that we develop rests on Itˆo’s formula and leads to p-th moment and pathwise α-exponential stability for p ≥ 2 and α > 0 with explicit Lyapunov exponents.

• Compagnoni, Enea; Orvieto, Antonio; Kersting, Hans; Proske, Frank Norbert & Lucchi, Aurelien (2024). SDEs for Minimax Optimization. Proceedings of Machine Learning Research (PMLR). 238, s. 4834–4842. Fulltekst i vitenarkiv Vis sammendrag

  Minimax optimization problems have attracted a lot of attention over the past few years, with applications ranging from economics to machine learning. While advanced optimization methods exist for such problems, characterizing their dynamics in stochastic scenarios remains notably challenging. In this paper, we pioneer the use of stochastic differential equations (SDEs) to analyze and compare Minimax optimizers. Our SDE models for Stochastic Gradient Descent-Ascent, Stochastic Extragradient, and Stochastic Hamiltonian Gradient Descent are provable approximations of their algorithmic counterparts, clearly showcasing the interplay between hyperparameters, implicit regularization, and implicit curvature-induced noise. This perspective also allows for a unified and simplified analysis strategy based on the principles of Itô calculus. Finally, our approach facilitates the derivation of convergence conditions and closed-form solutions for the dynamics in simplified settings, unveiling further insights into the behavior of different optimizers.

• Savku, Emel (2024). Deep-Control of Memory via Stochastic Optimal Control and Deep Learning. I Martinez, Viktor Gayoso; Queiruga-Dios, Araceli; Martin-Vaquero, Jesus; Yilmaz, Fatih; Rasteiro, Deolinda & Mierlus-Mazilu, Ion (Red.), Mathematical Methods for Engineering Applications ICMASE 2023, Madrid, Spain, July 12–14. Springer Nature. ISSN 9783031492174. s. 219–240. doi: 10.1007/978-3-031-49218-1_16. Fulltekst i vitenarkiv
• Font, Oriol Zamora; Baños, David & Ortiz-Latorre, Salvador (2024). Change of measure in a Heston–Hawkes stochastic volatility model. Advances in Applied Probability. ISSN 0001-8678. 56, s. 1370–1399. doi: 10.1017/apr.2024.14. Fulltekst i vitenarkiv Vis sammendrag

  We consider the stochastic volatility model obtained by adding a compound Hawkes process to the volatility of the well-known Heston model. A Hawkes process is a self-exciting counting process with many applications in mathematical finance, insurance, epidemiology, seismology, and other fields. We prove a general result on the existence of a family of equivalent (local) martingale measures. We apply this result to a particular example where the sizes of the jumps are exponentially distributed. Finally, a practical application to efficient computation of exposures is discussed.

• Agram, Nacira; Øksendal, Bernt Karsten & Rems, Jan (2024). Deep learning for quadratic hedging in incomplete jump market. Digital Finance. ISSN 2524-6984. 6(3), s. 463–499. doi: 10.1007/s42521-024-00112-5. Fulltekst i vitenarkiv Vis sammendrag

  We propose a deep learning approach to study the minimal variance pricing and hedging problem in an incomplete jump diffusion market. It is based on a rigorous stochastic calculus derivation of the optimal hedging portfolio, optimal option price, and the corresponding equivalent martingale measure through the means of the Stackelberg game approach. A deep learning algorithm based on the combination of the feed-forward and LSTM neural networks is tested on three different market models, two of which are incomplete. In contrast, the complete market Black–Scholes model serves as a benchmark for the algorithm’s performance. The results that indicate the algorithm’s good performance are presented and discussed. In particular, we apply our results to the special incomplete market model studied by Merton and give a detailed comparison between our results based on the minimal variance principle and the results obtained by Merton based on a different pricing principle. Using deep learning, we find that the minimal variance principle leads to typically higher option prices than those deduced from the Merton principle. On the other hand, the minimal variance principle leads to lower losses than the Merton principle.

• Sande, Åsmund Hausken & Wind, Johan Sokrates (2024). Minimal Convex Environmental Contours. SMAI Journal of Computational Mathematics (SMAI-JCM). 10, s. 55–83. doi: 10.5802/smai-jcm.106. Fulltekst i vitenarkiv
• Font, Oriol Zamora (2024). Pricing VIX options under the Heston-Hawkes stochastic volatility model. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv
• Agram, Nacira; Pucci, Giulia & Øksendal, Bernt Karsten (2024). Impulse Control of Conditional McKean–Vlasov Jump Diffusions. Journal of Optimization Theory and Applications. ISSN 0022-3239. 200, s. 1100–1130. doi: 10.1007/s10957-023-02370-6. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we consider impulse control problems involving conditional McKean–Vlasov jump diffusions, with the common noise coming from the σ-algebra generated by the first components of a Brownian motion and an independent compensated Poisson random measure. We first study the well-posedness of the conditional McKean–Vlasov stochastic differential equations (SDEs) with jumps. Then, we prove the associated Fokker–Planck stochastic partial differential equation (SPDE) with jumps. Next, we establish a verification theorem for impulse control problems involving conditional McKean–Vlasov jump diffusions. We obtain a Markovian system by combining the state equation with the associated Fokker–Planck SPDE for the conditional law of the state. Then we derive sufficient variational inequalities for a function to be the value function of the impulse control problem, and for an impulse control to be the optimal control. We illustrate our results by applying them to the study of an optimal stream of dividends under transaction costs. We obtain the solution explicitly by finding a function and an associated impulse control, which satisfy the verification theorem.

• Ren, Panpan; Tang, Hao & Wang, Feng-Yu (2024). Distribution-Path Dependent Nonlinear SPDEs with Application to Stochastic Transport Type Equations. Potential Analysis. ISSN 0926-2601. 61, s. 379–407. doi: 10.1007/s11118-023-10113-5. Fulltekst i vitenarkiv Vis sammendrag

  By using a regularity approximation argument, the global existence and uniqueness are derived for a class of nonlinear SPDEs depending on both the whole history and the distribution under strong enough noise. As applications, the global existence and uniqueness are proved for distribution-path dependent stochastic transport type equations, which are arising from stochastic fluid mechanics with forces depending on the history and the environment. In particular, the distribution-path dependent stochastic Camassa-Holm equation with or without Coriolis effect has a unique global solution when the noise is strong enough, whereas for the deterministic model wave-breaking may occur. This indicates that the noise may prevent blow-up almost surely.

• Benth, Fred Espen & Sgarra, Carlo (2024). A Barndorff-Nielsen and Shephard model with leverage in Hilbert space for commodity forward markets. Finance and Stochastics. ISSN 0949-2984. 28(4), s. 1035–1076. doi: 10.1007/s00780-024-00546-0. Fulltekst i vitenarkiv Vis sammendrag

  We propose an extension of the model introduced by Barndorff-Nielsen and Shephard, based on stochastic processes of Ornstein–Uhlenbeck type taking values in Hilbert spaces and including the leverage effect. We compute explicitly the characteristic function of the log-return and the volatility processes. By introducing a measure change of Esscher type, we provide a relation between the dynamics described with respect to the historical and the risk-neutral measures. We discuss in detail the application of the proposed model to describe the commodity forward curve dynamics in a Heath–Jarrow–Morton framework, including the modelling of forwards with delivery period occurring in energy markets and the pricing of options. For the latter, we show that a Fourier approach can be applied in this infinite-dimensional setting, relying on the attractive property of conditional Gaussianity of our stochastic volatility model. In our analysis, we study both arithmetic and geometric models of forward prices and provide appropriate martingale conditions in order to ensure arbitrage-free dynamics.

• Benth, Jurate Saltyte; Benth, Fred Espen & Nakstad, Espen Rostrup (2024). Nearly Instantaneous Time-Varying Reproduction Number for Contagious Diseases-a Direct Approach Based on Nonlinear Regression. Journal of Computational Biology. ISSN 1066-5277. 31(8), s. 727–741. doi: 10.1089/cmb.2023.0414. Fulltekst i vitenarkiv
• Nunno, Giulia Di & Yurchenko-Tytarenko, Anton (2024). Power law in Sandwiched Volterra Volatility model. Modern Stochastics: Theory and Applications (MSTA). ISSN 2351-6046. 11(2), s. 169–194. doi: 10.15559/24-VMSTA246. Fulltekst i vitenarkiv Vis sammendrag

  The paper presents an analytical proof demonstrating that the Sandwiched Volterra Volatility (SVV) model is able to reproduce the power-law behavior of the at-the-money implied volatility skew, provided the correct choice of the Volterra kernel. To obtain this result, the second-order Malliavin differentiability of the volatility process is assessed and the conditions that lead to explosive behavior in the Malliavin derivative are investigated. As a supplementary result, a general Malliavin product rule is proved.

• Mishura, Yuliya; Pilipenko, Andrey & Yurchenko-Tytarenko, Anton (2024). Low-dimensional Cox-Ingersoll-Ross process. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. 96(5), s. 1530–1550. doi: 10.1080/17442508.2023.2300291. Fulltekst i vitenarkiv Vis sammendrag

  The present paper investigates Cox-Ingersoll-Ross (CIR) processes of dimension less than 1, with a focus on obtaining an equation of a new type including local times for the square root of the CIR process. To derive this equation, we utilize the fact that non-negative diffusion processes can be obtained by the transformation of time and scale of a certain reflected Brownian motion. The equation mentioned above turns out to contain a term characterized by the local time of the corresponding reflected Brownian motion. Additionally, we establish a new connection between low-dimensional CIR processes and reflected Ornstein-Uhlenbeck (ROU) processes, providing a new representation of Skorokhod reflection functions.

• Benth, Fred Espen; Schroers, Dennis & Veraart, Almut E.D. (2024). A feasible central limit theorem for realised covariation of SPDEs in the context of functional data. The Annals of Applied Probability. ISSN 1050-5164. 34(2), s. 2208–2242. doi: 10.1214/23-AAP2019. Fulltekst i vitenarkiv
• Benth, Fred Espen; Lord, Gabriel J.; Nunno, Giulia Di & Petersson, Andreas Erik (2024). The heat modulated infinite dimensional Heston model and its numerical approximation. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. doi: 10.1080/17442508.2024.2424867. Fulltekst i vitenarkiv Vis sammendrag

  The HEat modulated Infinite DImensional Heston (HEIDIH) is introduced as a concrete case of the general framework of infinite dimensional Heston stochastic volatility models of (F.E. Benth, I.C. Simonsen '18) for the pricing of forward contracts. It is supported by a comprehensive numerical analysis. The model consists of a one-dimensional stochastic advection equation coupled with a stochastic volatility process. This process is a Cholesky-type decomposition of the tensor product of a Hilbert-space valued Ornstein-Uhlenbeck process, the solution to a stochastic heat equation on the real half-line. The advection and heat equations are driven by independent space-time Gaussian processes which are white in time and coloured in space, with the latter covariance structure expressed by two different kernels. First, a class of weight-stationary kernels are given, under which regularity results for the HEIDIH model in fractional Sobolev spaces are formulated. In particular, the class includes weighted Matérn kernels. Second, numerical approximation of the model is considered. An error decomposition formula, pointwise in space and time, for a finite-difference scheme is proven. For a special case, essentially sharp convergence rates are obtained when this is combined with a fully discrete finite element approximation of the stochastic heat equation. The analysis takes into account a localization error, a pointwise-in-space finite element discretization error and an error stemming from the noise being sampled pointwise in space. The rates obtained in the analysis are higher than what would be obtained using a standard Sobolev embedding technique. Numerical simulations illustrate the results.

• Bølviken, Erik Helmer & Haff, Ingrid Hobæk (2024). Loss modeling with many-parameter distributions. Scandinavian Actuarial Journal. ISSN 0346-1238. 2024(8), s. 763–780. doi: 10.1080/03461238.2024.2309987. Fulltekst i vitenarkiv Vis sammendrag

  It is argued that many-parameter families of loss distributions may work even with limited amounts of historical data. A restriction to unimodality works as a stabilizer, which makes fitted distributions much more stable than their parameters. We propose Box-Cox transformed Gamma and Burr variables. Those are models with three or four parameters with many of the traditional two-parameter families as special cases, and there are well-defined distributions at the boundaries of the parameter space, which is important for stability. The approach is evaluated with model error defined though the theory of misspecification in statistics. It is shown that such error is drastically reduced when a third or fourth parameter is added without increasing the random error more than a little. It is pointed out that the approach may be a suitable starting point for completely automatic procedures.

• Savku, Emel (2024). Memory and anticipation: two main theorems for Markov regime-switching stochastic processes. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. doi: 10.1080/17442508.2024.2427733. Fulltekst i vitenarkiv
• Benth, Fred Espen; Detering, Nils & Krühner, Paul (2024). Abstract polynomial processes. Electronic Journal of Probability (EJP). 29, s. 1–41. doi: 10.1214/24-EJP1182. Fulltekst i vitenarkiv Vis sammendrag

  We propose a novel approach to polynomial processes, which allows us to analyse such processes on general state spaces. We do not need to specify polynomials explicitly but can work with a general sequence of graded vector spaces of functions on the state space. Elements of these graded vector spaces form the polynomials. By introducing a sequence of vector space complements, we obtain the sets of monomials. The basic tool of our analysis is the polynomial action operator, which is a semigroup of operators mapping conditional expected values of polynomials acting on a polynomial process to polynomials of the same or lower grade. We study abstract polynomial processes under algebraic and topological assumptions on the polynomial actions, and establish an affine drift structure. Moreover, we characterize the covariance structure under similar but slightly stronger conditions. A crucial part in our analysis is the use of the (algebraic or topological) dual of the monomials of grade one, which serves as a linearization of the state space of the polynomial process. We provide several examples of polynomial processes that do not fall into the classical setting but are polynomial processes according to our definition and can be analyzed with the tools we provide here.

• Agram, Nacira & Øksendal, Bernt Karsten (2024). Optimal stopping of conditional McKean–Vlasov jump diffusions. Systems & control letters (Print). ISSN 0167-6911. 188. doi: 10.1016/j.sysconle.2024.105815. Fulltekst i vitenarkiv Vis sammendrag

  The purpose of this paper is to study the optimal stopping problem of conditional McKean–Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean–Vlasov jump diffusions, for short). We obtain sufficient variational inequalities for a function to be the value function of such a problem and for a stopping time to be optimal. The key is that we combine the conditional McKean–Vlasov equation with the associated stochastic Fokker–Planck partial integro-differential equation for the conditional law of the state. This leads to a Markovian system which can be handled by using a version of a Dynkin formula. Our verification result is illustrated by finding the optimal time to sell in a market with common noise and jumps.

• Agram, Nacira; Øksendal, Bernt Karsten; Proske, Frank Norbert & Tymoshenko, Olena (2024). Fokker-Planck equation for McKean-Vlasov SPDEs driven by time-space Brownian sheet. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we consider a McKean-Vlasov (mean-field) stochastic partial differential equations (SPDEs) driven by a Brownian sheet. We study the propagation of chaos for a space-time Ornstein-Uhlenbeck SPDE type. Subsequently, we prove the existence and uniqueness of a nonlinear McKean-Vlasov SPDE. Finally, we establish a Fokker-Planck equation for the law of the solution of the McKean-Vlasov type SPDE driven by a time-space Brownian sheet, and we provide some examples to illustrate the results obtained.

• Grochowicz, Aleksander; Greevenbroek, Koen van & Bloomfield, Hannah C. (2024). Using power system modelling outputs to identify weather-induced extreme events in highly renewable systems. Environmental Research Letters. 19(5). doi: 10.1088/1748-9326/ad374a. Fulltekst i vitenarkiv Vis sammendrag

  In highly renewable power systems the increased weather dependence can result in new resilience challenges, such as renewable energy droughts, or a lack of sufficient renewable generation at times of high demand. The weather conditions responsible for these challenges have been well-studied in the literature. However, in reality multi-day resilience challenges are triggered by complex interactions between high demand, low renewable availability, electricity transmission constraints and storage dynamics. We show these challenges cannot be rigorously understood from an exclusively power systems, or meteorological, perspective. We propose a new method that uses electricity shadow prices—obtained by a European power system model based on 40 years of reanalysis data—to identify the most difficult periods driving system investments. Such difficult periods are driven by large-scale weather conditions such as low wind and cold temperature periods of various lengths associated with stationary high pressure over Europe. However, purely meteorological approaches fail to identify which events lead to the largest system stress over the multi-decadal study period due to the influence of subtle transmission bottlenecks and storage issues across multiple regions. These extreme events also do not relate strongly to traditional weather patterns (such as Euro-Atlantic weather regimes or the North Atlantic Oscillation index). We therefore compile a new set of weather patterns to define energy system stress events which include the impacts of electricity storage and large-scale interconnection. Without interdisciplinary studies combining state-of-the-art energy meteorology and modelling, further strive for adequate renewable power systems will be hampered.

• Amine, Oussama; Baños, David & Proske, Frank Norbert (2024). C∞-Regularization by Noise of Singular ODE’s. Journal of Dynamics and Differential Equations. ISSN 1040-7294. 36, s. 915–974. doi: 10.1007/s10884-024-10355-w. Fulltekst i vitenarkiv Vis sammendrag

  In this paper we construct a new type of noise of fractional nature that has a strong regularizing effect on differential equations. We consider an equation driven by a highly irregular vector field and study the effect of this noise on such dynamical systems. We employ a new method to prove existence and uniqueness of global strong solutions, where classical methods fail because of the “roughness” and non-Markovianity of the driving process. In addition, we prove the rather remarkable property that such solutions are infinitely many times classically differentiable with respect to the initial condition in spite of the vector field being discontinuous. The technique used in this article corresponds, in a certain sense, to the Nash–Moser iterative scheme in combination with a new concept of “higher order averaging operators along highly fractal stochastic curves”. This approach may provide a general principle for the study of regularization by noise effects in connection with important classes of partial differential equations.

• Lozano, Pere Diaz; Bagén, Toni Lozano & Vives, Josep (2023). Neural stochastic differential equations for conditional time series generation using the Signature-Wasserstein-1 metric. Journal of Computational Finance. ISSN 1460-1559. 27(1), s. 1–23. doi: 10.21314/jcf.2023.005. Fulltekst i vitenarkiv Vis sammendrag

  (Conditional) generative adversarial networks (GANs) have had great success in recent years, due to their ability to approximate (conditional) distributions over extremely high-dimensional spaces. However, they are highly unstable and computationally expensive to train, especially in the time series setting. Recently, the use of a key object in rough path theory, called the signature of a path, has been proposed. This is able to convert the min–max formulation given by the (conditional) GAN framework into a classical minimization problem. However, this method is extremely costly in terms of memory, which can sometimes become prohibitive. To overcome this, we propose the use of conditional neural stochastic differential equations, designed to have a constant memory cost as a function of depth, being more memory efficient than traditional deep learning architectures. We empirically test the efficiency of our proposed model against other classical approaches, in terms of both memory cost and computational time, and show that it usually outperforms them according to several metrics.

• Puica, Mihaela-Alexandra & Benth, Fred Espen (2023). A spatio-temporal model for predicting wind speeds in Southern California. Communications in statistics. Case studies, data analysis and applications.. ISSN 2373-7484. 9(3), s. 321–349. doi: 10.1080/23737484.2023.2217137. Fulltekst i vitenarkiv
• Hachemi, Rahma Yasmina Moulay & Øksendal, Bernt Karsten (2023). The fractional stochastic heat equation driven by time-space white noise. Fractional Calculus and Applied Analysis. ISSN 1311-0454. 26(2), s. 513–532. doi: 10.1007/s13540-023-00134-7. Fulltekst i vitenarkiv
• Grochowicz, Aleksander; Greevenbroek, Koen van; Benth, Fred Espen & Zeyringer, Marianne (2023). Intersecting near-optimal spaces: European power systems with more resilience to weather variability. Energy Economics. ISSN 0140-9883. 118. doi: 10.1016/j.eneco.2022.106496. Fulltekst i vitenarkiv Vis sammendrag

  We suggest a new methodology for designing robust energy systems. For this, we investigate so-called near-optimal solutions to energy system optimisation models; solutions whose objective values deviate only marginally from the optimum. Using a refined method for obtaining explicit geometric descriptions of these near-optimal feasible spaces, we find designs that are as robust as possible to perturbations. This contributes to the ongoing debate on how to define and work with robustness in energy systems modelling. We apply our methods in an investigation using multiple decades of weather data. For the first time, we run a capacity expansion model of the European power system (one node per country) with a three-hourly temporal resolution and 41 years of weather data. While an optimisation with 41 weather years is at the limits of computational feasibility, we use the near-optimal feasible spaces of single years to gain an understanding of the design space over the full time period. Specifically, we intersect all near-optimal feasible spaces for the individual years in order to get designs that are likely to be feasible over the entire time period. We find significant potential for investment flexibility, and verify the feasibility of these designs by simulating the resulting dispatch problem with four decades of weather data. They are characterised by a shift towards more onshore wind and solar power, while emitting more than 50% less CO2 than a cost-optimal solution over that period. Our work builds on recent developments in the field, including techniques such as Modelling to Generate Alternatives (MGA) and Modelling All Alternatives (MAA), and provides new insights into the geometry of near-optimal feasible spaces and the importance of multi-decade weather variability for energy systems design. We also provide an effective way of working with a multi-decade time frame in a highly parallelised manner. Our implementation is open-sourced, adaptable and is based on PyPSA-Eur.

• Giordano, Michele & Yurchenko-Tytarenko, Anton (2023). Optimal control in linear-quadratic stochastic advertising models with memory. Decisions in Economics and Finance (DAF). ISSN 1593-8883. doi: 10.1007/s10203-023-00409-x. Fulltekst i vitenarkiv Vis sammendrag

  Abstract This paper deals with a class of optimal control problems which arises in advertising models with Volterra Ornstein-Uhlenbeck process representing the product goodwill. Such choice of the model can be regarded as a stochastic modification of the classical Nerlove-Arrow model that allows to incorporate both presence of uncertainty and empirically observed memory effects such as carryover or distributed forgetting. We present an approach to solve such optimal control problems based on an infinite dimensional lift which allows us to recover Markov properties by formulating an optimization problem equivalent to the original one in a Hilbert space. Such technique, however, requires the Volterra kernel from the forward equation to have a representation of a particular form that may be challenging to obtain in practice. We overcome this issue for Hölder continuous kernels by approximating them with Bernstein polynomials, which turn out to enjoy a simple representation of the required type. Then we solve the optimal control problem for the forward process with approximated kernel instead of the original one and study convergence. The approach is illustrated with simulations.

• Huseby, Arne Bang (2023). Environmental contours and time dependence. I Brito, Mario P.; Aven, Terje; Baraldi, Piero; Cepin, Marko & Zio, Enrico (Red.), ESREL 2023 - Proceedings of the 33rd European Safety and Reliability Conference : The Future of Safety in the Reconnected World, 3 – 7 September 2023, University of Southampton, United Kingdom. Research Publishing Services. ISSN 9789811880711. s. 1295–1302. doi: 10.3850/978-981-18-8071-1_P048-cd. Fulltekst i vitenarkiv Vis sammendrag

  Environmental contours are widely used as a basis for e.g., ship design. Such contours are typically used in early design when the strength and failure properties of the object under consideration are not known. An environmental contour describes the tail properties of some relevant environmental variables, and is used as input to the design process. A methodology for constructing environmental contours based on the Rosenblatt transformation has beed used extensively. More recently alternative approach where environmental contours are constructed using Monte Carlo simulation have been developed. Typically, the strength of a structural design is chosen so that the expected return period of a failure event exceeds the desired lifetime of the structure. If time dependence in the environmental variables is neglected, the expected return period is simply the inverse of the failure probability. In a more realistic model, however, such dependence should be included. In this paper we describe a method for constructing an environmental contour where time dependence is taken into account. The method is illustrated with a simple numerical example.

• Banos, David Ruiz; Sande, Åsmund Hausken & Sgarra, Carlo (2023). Guaranteed Minimum Maturity Benefits in a Self-Exciting Stochastic Mortality Model: Pricing, Estimation and Calibration. North American Actuarial Journal (NAAJ). ISSN 1092-0277. 28(4), s. 744–771. doi: 10.1080/10920277.2023.2254836. Fulltekst i vitenarkiv Vis sammendrag

  The guaranteed minimum maturity benefit (GMMB) is quite a popular feature embedded in several unit-linked policies offered by insurance companies. The value of this benefit depends on several processes assumed to describe both the mortality and the financial dynamics, typically represented by interest rates and by the fund associated to the unit-linked policy. A large literature is devoted to the valuation of GMMB for different mortality models, in particular when the mortality dynamics is described by affine models of diffusion type. In the present article we assume a self-exciting behavior for the mortality dynamics, described by a Hawkes-type process with exponential kernel. This allows us to keep both the Markov and affine features but introduces jumps with a stochastic intensity. These types of dynamics, exhibiting a jump clustering property, are quite convenient for describing mortality in some critical situations, like epidemics, where contagion phenomena make the probability of jump arrivals higher whenever a jump occurs. By assuming diffusion-type dynamics for both the fund and the interest rate and introducing all of the possible correlations among the diffusion processes necessary to obtain realistic dynamics, we take advantage of the affine features of the model proposed and compute in a semi-explicit form the value of a GMMB. Lastly, we calibrate our model and perform an empirical analysis to determine the effects of excess mortality on GMMB prices as well as a death benefit portfolio.

• Sande, Åsmund Hausken (2023). Convex environmental contours for non-stationary processes. Ocean Engineering. ISSN 0029-8018. 292. doi: 10.1016/j.oceaneng.2023.116615. Fulltekst i vitenarkiv Vis sammendrag

  Environmental contours are tools frequently used in the early design of marine structures. They provide a description of critical design conditions and serve as a means for simplifying expensive long-term response calculations. Here, we consider convex contours based on the assumption of convex failure sets. We provide a rigorous foundation for the existence of such contours when the underlying environmental factors are modelled by a general, possibly non-stationary, process. This constitutes a generalisation of existing theory and is done to properly account for empirically observed increases in extreme sea-states. Two definitions are proposed, based respectively on averages or quantiles of exceedence times, along with minimal conditions on the environmental processes to guarantee existence. In order to illustrate these methods we give two examples, including an empirical study containing a method for constructing contours based on the presented theory.

• Aman, Auguste; Coulibaly, Harouna & Dordevic, Jasmina (2023). Forward-backward stochastic differential equations with delay generators. Stochastics and Dynamics. ISSN 0219-4937. 23(02). doi: 10.1142/S0219493723500120. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we prove a result of existence and uniqueness of solutions to coupled forward–backward stochastic differential equations with delayed generators under a Lipschitz condition.

• Agram, Nacira & Øksendal, Bernt Karsten (2023). Stochastic Fokker–Planck Equations for Conditional McKean–Vlasov Jump Diffusions and Applications to Optimal Control. SIAM Journal of Control and Optimization. ISSN 0363-0129. 61(3), s. 1472–1493. doi: 10.1137/21M1461034. Fulltekst i vitenarkiv Vis sammendrag

  The purpose of this paper is to study optimal control of conditional McKean–Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean–Vlasov jump diffusions, for short). To this end, we first prove a stochastic Fokker–Planck equation for the conditional law of the solution of such equations. Combining this equation with the original state equation, we obtain a Markovian system for the state and its conditional law. Furthermore, we apply this to formulate a Hamilton–Jacobi–Bellman equation for the optimal control of conditional McKean–Vlasov jump diffusions. Then we study the situation when the law is absolutely continuous with respect to Lebesgue measure. In that case the Fokker–Planck equation reduces to a stochastic partial differential equation for the Radon–Nikodym derivative of the conditional law. Finally we apply these results to solve explicitly the linear-quadratic optimal control problem of conditional stochastic McKean–Vlasov jump diffusions, and optimal consumption from a cash flow.

• Benth, Fred Espen & Lempa, Jukka (2023). Hedging temperature risk with CDD and HDD temperature futures. Applied Stochastic Models in Business and Industry. ISSN 1524-1904. doi: 10.1002/asmb.2815. Fulltekst i vitenarkiv Vis sammendrag

  Abstract This paper is concerned with managing risk exposure to temperature using weather derivatives. We consider hedging temperature risk using so‐called HDD‐ and CDD‐index futures, which are instruments written on temperatures in specific locations over specific time periods. The temperatures are modelled as continuous‐time autoregressive (CARMA) processes and pricing of the hedging instrument is done under an equivalent pricing measure. We develop hedging strategies for locations, cutoff temperatures, and time periods different to the ones in the traded contracts, allowing for more flexibility in the hedging application. The dynamic hedging strategies are expressed explicitly by the term structure of the volatility. We also provide numerical case studies with temperatures following a CAR(3)‐process to illustrate the temporal behaviour of the hedge under different scenarios.

• Baños, David; Bauer, Martin; Meyer-Brandis, Thilo & Proske, Frank Norbert (2023). Restoration of Well-Posedness of Infinite-Dimensional Singular ODE’s via Noise. Potential Analysis. ISSN 0926-2601. doi: 10.1007/s11118-023-10069-6. Fulltekst i vitenarkiv Vis sammendrag

  Abstract In this paper we aim at generalizing the results of A. K. Zvonkin ( That removes the drift , 22 , 129, 41) and A. Y. Veretennikov ( Theory Probab. Appl. , 24 , 354, 39) on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the Euclidean space to the case of infinite-dimensional state spaces. The regularizing driving noise in our equation is chosen to be a locally non-Hölder continuous Hilbert space valued process of fractal nature, which does not allow for the use of classical construction techniques for strong solutions from PDE or semimartingale theory. Our approach, which does not resort to the Yamada-Watanabe principle for the verification of pathwise uniqueness of solutions, is based on Malliavin calculus.

• Tang, Hao & Wang, Zhi-An (2023). Strong solutions to a nonlinear stochastic aggregation-diffusion equation. Communications in Contemporary Mathematics. ISSN 0219-1997. doi: 10.1142/S0219199722500730. Fulltekst i vitenarkiv
• Coffie, Emmanuel; Mao, Xuerong & Proske, Frank Norbert (2023). On the Analysis of Ait-Sahalia-Type Model for Rough Volatility Modelling. Journal of theoretical probability. ISSN 0894-9840. doi: 10.1007/s10959-023-01269-2. Fulltekst i vitenarkiv
• Savku, Emel (2023). A Stochastic Control Approach for Constrained Stochastic Differential Games with Jumps and Regimes. Mathematics. 11(14). doi: 10.3390/math11143043. Fulltekst i vitenarkiv Vis sammendrag

  We develop an approach for two-player constraint zero-sum and nonzero-sum stochastic differential games, which are modeled by Markov regime-switching jump-diffusion processes. We provide the relations between a usual stochastic optimal control setting and a Lagrangian method. In this context, we prove corresponding theorems for two different types of constraints, which lead us to find real-valued and stochastic Lagrange multipliers, respectively. Then, we illustrate our results for a nonzero-sum game problem with the stochastic maximum principle technique. Our application is an example of cooperation between a bank and an insurance company, which is a popular, well-known business agreement type called Bancassurance.

• Benth, Fred Espen; Detering, Nils & Galimberti, Luca (2023). Pricing options on flow forwards by neural networks in a Hilbert space. Finance and Stochastics. ISSN 0949-2984. 28, s. 81–121. doi: 10.1007/s00780-023-00520-2. Fulltekst i vitenarkiv
• Tang, Hao (2023). On the stochastic Euler-Poincaré equations driven by pseudo-differential/multiplicative noise. Journal of Functional Analysis. ISSN 0022-1236. 285(9). doi: 10.1016/j.jfa.2023.110075. Fulltekst i vitenarkiv Vis sammendrag

  The stochastic Euler-Poincaré equations with pseudo-differential/multiplicative noise are considered in this work. We first establish two new cancellation properties on pseudo-differential operators, which considerably extend the previous results for transport type noise only involving gradient operator. Then, we obtain results on local solution, blow-up criterion, and global existence. The interplay between stability on exiting times and continuous dependence of solution on initial data is also studied for the multiplicative noise case.

• Amine, Oussama; Mansouri, Abdol-Reza & Proske, Frank Norbert (2023). Well-posedness of the deterministic transport equation with singular velocity field perturbed along fractional Brownian paths. Journal of Differential Equations. ISSN 0022-0396. 362, s. 106–172. doi: 10.1016/j.jde.2023.02.059. Fulltekst i vitenarkiv
• Benth, Fred Espen & Karbach, Sven (2023). Multivariate continuous-time autoregressive moving-average processes on cones. Stochastic Processes and their Applications. ISSN 0304-4149. 162, s. 299–337. doi: 10.1016/j.spa.2023.05.003. Fulltekst i vitenarkiv
• Bordin, Chiara; Mishra, Sambeet & Benth, Fred Espen (2023). Pedagogical Perspectives of Interdisciplinary Teaching and Research: An Energy System Modelling Outlook in Relation to Energy Informatics. Energies. 16(15). doi: 10.3390/en16155757. Fulltekst i vitenarkiv Vis sammendrag

  The purpose of this paper is to present and discuss pedagogical frameworks and approaches to developing, delivering, and evaluating a new interdisciplinary course within the domain of energy informatics at both Master’s and PhD levels. This study is needed because many papers on sustainable energy engineering education concentrate on course content but provide very little information on the pedagogical methods employed to deliver that content. The proposed new course is called “smart energy and power systems modelling” and is aimed at discussing how mathematical optimization, in the context of computer science, can contribute to more effectively managing smart energy and power systems. Different pedagogical frameworks are discussed and adapted for the specific domain of energy informatics. An ASSURE model coupled with Bloom’s taxonomy is presented for the design of the course and identification of learning objectives; self-regulated learning strategies are discussed to enhance the learning process; a novel model called GPD (Gaussian Progression of Difficulty) for lecture planning was proposed; a teaching-research nexus is discussed for the course planning and enhancement. Adopting qualitative analyses and an inductive approach, this paper offers a thorough reflection on the strengths and weaknesses of the new course, together with improvement possibilities based on fieldwork and direct experience with the students and colleagues. Opportunities and challenges of interdisciplinary teaching are presented in light of real-world experience, with a particular focus on the interaction between mathematics and computer science to study the specific application of energy and power systems.

• Agram, Nacira & Øksendal, Bernt Karsten (2023). The Donsker delta function and local time for McKean–Vlasov processes and applications. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. doi: 10.1080/17442508.2023.2286252. Fulltekst i vitenarkiv Vis sammendrag

  The purpose of this paper is to establish a stochastic differential equation for the Donsker delta measure of the solution of a McKean–Vlasov (mean-field) stochastic differential equation. If the Donsker delta measure is absolutely continuous with respect to Lebesgue measure, then its Radon–Nikodym derivative is called the Donsker delta function. In that case it can be proved that the local time of such a process is simply the integral with respect to time of the Donsker delta function. Therefore we also get an equation for the local time of such a process. For some particular McKean–Vlasov processes, we find explicit expressions for their Donsker delta functions and hence for their local times.

• Larsson, Karl; Green, Rikard & Benth, Fred Espen (2023). A stochastic time-series model for solar irradiation. Energy Economics. ISSN 0140-9883. 117. doi: 10.1016/j.eneco.2022.106421. Fulltekst i vitenarkiv Vis sammendrag

  We propose a novel stochastic time series model able to explain the stylized features of daily irradiation level data in 5 cities in Germany. The model is suitable for applications to risk management of photovoltaic power production in renewable energy markets. The suggested dynamics is a low-order autoregressive time series with seasonal level given by an atmospheric clear-sky model. Moreover, we detect a skewness property in the residuals which we explain by a winter–summer regime switch. The stochastic variance is modeled by a seasonally varying GARCH-dynamics. The winter and summer standardized residuals are proposed to be a Gaussian mixture model to capture the bimodal distributions. We estimate the model on the observed data, and perform a validation study. An application to energy markets studying the production at risk for a PV-producer is presented.

• Benth, Fred Espen; Deelstra, Griselda & Kozplnar, And Sinem (2023). Pricing energy quanto options in the framework of Markov-modulated additive processes. IMA Journal of Management Mathematics. ISSN 1471-678X. 34(1), s. 187–220. doi: 10.1093/imaman/dpab032. Fulltekst i vitenarkiv
• Makhlouf, K.; Agram, Nacira; Hilbert, A. & Øksendal, Bernt Karsten (2023). SPDEs with space interactions and application to population modelling. ESAIM: Control, Optimisation and Calculus of Variations (ESAIM: COCV). ISSN 1292-8119. 29. doi: 10.1051/cocv/2023010. Fulltekst i vitenarkiv Vis sammendrag

  We consider optimal control of a new type of non-local stochastic partial differential equations (SPDEs). The SPDEs have space interactions , in the sense that the dynamics of the system at time t and position in space x also depend on the space-mean of values at neighbouring points. This is a model with many applications, e.g. to population growth studies and epidemiology. We prove the existence and uniqueness of strong, smooth solutions of a class of SPDEs with space interactions, and we show that, under some conditions, the solutions are positive for all times if the initial values are. Sufficient and necessary maximum principles for the optimal control of such systems are derived. Finally, we apply the results to study an optimal vaccine strategy problem for an epidemic by modelling the population density as a space-mean stochastic reaction-diffusion equation.

• Dordevic, Jasmina & Jovanović, Bojana (2023). Dynamical analysis of a stochastic delayed epidemic model with lévy jumps and regime switching. Journal of the Franklin Institute. ISSN 0016-0032. 360(2), s. 1252–1283. doi: 10.1016/j.jfranklin.2022.12.009. Fulltekst i vitenarkiv Vis sammendrag

  In this paper a delayed stochastic SLVIQR epidemic model, which can be applied for modeling the new coronavirus COVID-19 after a calibration, is derived. Model is constructed by assuming that transmission rate satisfies the mean-reverting Ornstain-Uhlenbeck process and, besides a standard Brownian motion, another two driving processes are considered: a stationary Poisson point process and a continuous finite-state Markov chain. For the constructed model, the existence and uniqueness of positive global solution is proven. Also, sufficient conditions under which the disease would lead to extinction or be persistent in the mean are established and it is shown that constructed model has a richer dynamic analysis compared to existing models. In addition, numerical simulations are given to illustrate the theoretical results.

• Lindstrøm, Tom Louis (2023). The Allure of Infinitesimals: Sergio Albeverio and Nonstandard Analysis. I Hilbert, Astrid; Mastrogiacomo, Elisa; Mazzucchi, Sonia; Rüdiger, Barbara & Ugolini, Stefania (Red.), Quantum and Stochastic Mathematical Physics. Springer. ISSN 9783031140310. s. 187–215. doi: 10.1007/978-3-031-14031-0_9. Fulltekst i vitenarkiv Vis sammendrag

  I give a survey of Sergio Albeverio’s work in nonstandard analysis, covering applications to operator theory, stochastic analysis, Dirichlet forms, quantum mechanics, and quantum field theory, and making an attempt at putting his contributions into the historical context of what has happened in the field before and since.

• Catellier, Rémi & Harang, Fabian Andsem (2023). Pathwise regularization of the stochastic heat equation with multiplicative noise through irregular perturbation. Annales de l'I.H.P. Probabilites et statistiques. ISSN 0246-0203. 59(3), s. 1572–1609. doi: 10.1214/22-AIHP1302. Fulltekst i vitenarkiv Vis sammendrag

  Existence and uniqueness of solutions to the stochastic heat equation with multiplicative spatial noise is studied. In the spirit of pathwise regularization by noise, we show that a perturbation by a sufficiently irregular continuous path establish wellposedness of such equations, even when the drift and diffusion coefficients are given as generalized functions or distributions. In addition we prove regularity of the averaged field associated to a Lévy fractional stable motion, and use this as an example of a perturbation regularizing the multiplicative stochastic heat equation.

• Kersting, Hans; Orvieto, Antonio; Proske, Frank Norbert & Lucchi, Aurelien (2023). Mean first exit times of Ornstein-Uhlenbeck processes in high-dimensional spaces. Journal of Physics A: Mathematical and Theoretical. ISSN 1751-8113. 56(21). doi: 10.1088/1751-8121/acc559. Fulltekst i vitenarkiv
• Dordevic, Jasmina (2023). Backward Doubly Stochastic Integral Equations of the Volterra Type and Some Related Problems. Communications in Mathematics and Statistics. ISSN 2194-6701. doi: 10.1007/s40304-023-00349-3. Fulltekst i vitenarkiv Vis sammendrag

  Backward doubly stochastic integral equations of the Volterra type (BDSIEVs in short) are observed in this paper. Existence of M-solution established under functional Lipschitz assumptions. Duality principle between linear BDSIEVs and (forward) stochastic Volterra integral equations is obtained. Using duality principle, the comparison theorem for the adapted solutions of BDSIEVs is proven.

• Nunno, Giulia Di; Ortiz-Latorre, Salvador & Petersson, Andreas Erik (2023). SPDE bridges with observation noise and their spatial approximation. Stochastic Processes and their Applications. ISSN 0304-4149. 158, s. 170–207. doi: 10.1016/j.spa.2023.01.007. Fulltekst i vitenarkiv Vis sammendrag

  This paper introduces SPDE bridges with observation noise and contains an analysis of their spatially semidiscrete approximations. The SPDEs are considered in the form of mild solutions in an abstract Hilbert space framework suitable for parabolic equations. They are assumed to be linear with additive noise in the form of a cylindrical Wiener process. The observational noise is also cylindrical and SPDE bridges are formulated via conditional distributions of Gaussian random variables in Hilbert spaces. A general framework for the spatial discretization of these bridge processes is introduced. Explicit convergence rates are derived for a spectral and a finite element based method. It is shown that for sufficiently rough observation noise, the rates are essentially the same as those of the corresponding discretization of the original SPDE.

• Bogso, Antoine-Marie; Menoukeu-Pamen, Olivier & Proske, Frank Norbert (2023). EXISTENCE OF STRONG SOLUTIONS OF FRACTIONAL BROWNIAN SHEET DRIVEN SDES WITH INTEGRABLE DRIFT. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv
• Compagnoni, Enea; Biggio, Luca; Orvieto, Antonio; Proske, Frank Norbert; Kersting, Hans & Lucchi, Aurelien (2023). An SDE for Modeling SAM: Theory and Insights. Proceedings of Machine Learning Research (PMLR). 202, s. 25209–25253. Fulltekst i vitenarkiv Vis sammendrag

  We study the SAM (Sharpness-Aware Minimization) optimizer which has recently attracted a lot of interest due to its increased performance over more classical variants of stochastic gradient descent. Our main contribution is the derivation of continuous-time models (in the form of SDEs) for SAM and two of its variants, both for the fullbatch and mini-batch settings. We demonstrate that these SDEs are rigorous approximations of the real discrete-time algorithms (in a weak sense, scaling linearly with the learning rate). Using these models, we then offer an explanation of why SAM prefers flat minima over sharp ones – by showing that it minimizes an implicitly regularized loss with a Hessian-dependent noise structure. Finally, we prove that SAM is attracted to saddle points under some realistic conditions. Our theoretical results are supported by detailed experiments.

• Puica, Mihaela-Alexandra; Dehant, V.; Folgueira, M.; Hoolst, Hoolst Van & Rekier, J. (2023). Analytical computation of total topographic torque at the core-mantle boundary and its impact on tidally driven length-of-day variations. Geophysical Journal International. ISSN 0956-540X. 234(1), s. 585–596. doi: 10.1093/gji/ggad077. Fulltekst i vitenarkiv Vis sammendrag

  SUMMARY The Earth’s rotation exhibits periodic variations as a result of gravitational torques exerted by the Sun and the Moon and of angular momentum exchange of the solid Earth with the Earth’s atmosphere and hydrosphere. Here, we aim at determining the complementary effect of the deep interior on variations in the length-of-day (LOD) and focus on the influence of topography at the core–mantle boundary (CMB). For this purpose, we have developed an analytical approach for solving the Navier–Stokes equation for global rotational motions and inertial waves, based on and extending the approach of Wu & Wahr (1997). An advantage of the analytical approach is that it allows to identify the frequencies and topographic spherical harmonics degrees and orders where resonance can happen, as well as to quantify the total amplifications in the tidal effects on LOD variations. Although the resonances are found to be sometimes quite near tidal frequencies, we show that they are not sufficiently close to induce significant perturbations in LOD variations, except for two of the tides, the fortnightly and monthly tides Mf and Mm. Our results go beyond the findings of Wu & Wahr (1997), extending them to a much wider range of degrees and orders of topographic coefficients. We show that there is an amplification in Mf and Mm induced by the degree 18-order 10 and by the degree 7-order 1 of the topography, respectively. Our approach is generic in the sense that it can be applied to other orientation changes of the Earth as well as to other planets.

• Font, Oriol Zamora; Baños, David & Ortiz-Latorre, Salvador (2023). Thiele's PIDE for unit-linked policies in the Heston-Hawkes stochastic volatility model. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv
• Lindstrøm, Tom Louis & Ross, David Andrew (2023). A nonstandard approach to asymptotic fixed point theorems. Journal of Fixed Point Theory and Applications. ISSN 1661-7738. 25(1). doi: 10.1007/s11784-022-01028-6. Fulltekst i vitenarkiv Vis sammendrag

  We use methods from nonstandard analysis to unify and extend results from the theory of asymptotic fixed points on metric spaces. Among the topics we consider are Kirk’s asymptotic fixed point theorem with extensions, Boyd and Wong’s fixed point theorem, and Suzuki’s fixed point theorem for asymptotic contractions of the Meier–Keeler type. We also show how Suzuki’s asymptotic contractions of the final type can be modified to yield uniform convergence on compact and bounded sets.

• Agrell, Christian; Dahl, Kristina Rognlien & Hafver, Andreas (2023). Optimal sequential decision making with probabilistic digital twins. SN Applied Sciences. ISSN 2523-3963. 5(4). doi: 10.1007/s42452-023-05316-9. Fulltekst i vitenarkiv Vis sammendrag

  In this study, we present a formal defnition of the probabilistic digital twin (PDT). Digital twins are emerging in many industries, typically consisting of simulation models and data associated with a specifc physical system. In order to defne probabilistic digital twins, we discuss how epistemic uncertainty can be treated using measure theory, by modelling epistemic information via sigma-algebras. A gentle introduction to the necessary mathematical theory is provided throughout the paper, together with a number of examples to illustrate the core concepts. We then introduce the problem of optimal sequential decision making. That is, when the outcome of each decision may inform the next. We discuss how this problem may be solved theoretically, and the current limitations that prohibit most practical applications. As a numerically tractable alternative, we propose a generic approximate solution using deep reinforcement learning together with neural networks defined on sets. We illustrate the method on a practical problem, considering optimal information gathering for the estimation of a failure probability.

• Øksendal, Bernt Karsten (2023). Space-time stochastic calculus and white noise. I Morel, Jean-Michel & Teissier, Bernard (Red.), Mathematics Going Forward - Collected Mathematical Brushstrokes. Springer. ISSN 9783031122439. s. 629–649. doi: 10.1007/978-3-031-12244-6_44. Fulltekst i vitenarkiv Vis sammendrag

  In the first part of this paper I give the historical background to my initial interest in stochastic analysis and to the writing of my book Stochastic Differential Equations. The first edition of this book was published by Springer in 1985, with the highly appreciated support of Catriona Byrne. In the second part I present a motivation for modelling the dynamics of a system subject to a noise by means of a stochastic partial differential equation (SPDE) driven by a time-space Brownian sheet. This is followed by a brief survey of time-space white noise and Hida–Malliavin calculus, which are useful tools for studying such equations. As an illustration I apply white noise calculus to find an explicit solution of an SPDE describing population growth in an environment subject to time-space white noise. Dedicated to Catriona Byrne, in gratitude for her support and encouragement through 40 years.

• Nunno, Giulia Di & Giordano, Michele (2023). Stochastic Volterra equations with time-changed Lévy noise and maximum principles. Annals of Operations Research. ISSN 0254-5330. doi: 10.1007/s10479-023-05303-8. Fulltekst i vitenarkiv Vis sammendrag

  Motivated by a problem of optimal harvesting of natural resources, we study a control problem for Volterra type dynamics driven by time-changed Lévy noises, which are in general not Markovian. To exploit the nature of the noise, we make use of different kind of information flows within a maximum principle approach. For this we work with backward stochastic differential equations (BSDE) with time-change and exploit the non-anticipating stochastic derivative introduced in Di Nunno and Eide (Stoch Anal Appl 28:54-85, 2009). We prove both a sufficient and necessary stochastic maximum principle.

• Mishura, Yuliya & Yurchenko-Tytarenko, Anton (2023). Parameter Estimation in Rough Bessel Model. Fractal and Fractional. 7(7). doi: 10.3390/fractalfract7070508. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we construct consistent statistical estimators of the Hurst index, volatility coefficient, and drift parameter for Bessel processes driven by fractional Brownian motion with H<1/2. As an auxiliary result, we also prove the continuity of the fractional Bessel process. The results are illustrated with simulations.

• Amine, Oussama; Baghery, Karim; Pindado, Zaira & Ràfols, Carla (2023). Simulation extractable versions of Groth’s zk-SNARK revisited. International Journal of Information Security. ISSN 1615-5262. 23, s. 431–445. doi: 10.1007/s10207-023-00750-7. Fulltekst i vitenarkiv Vis sammendrag

  Zero-knowledge succinct non-interactive arguments of knowledge (zk-SNARKs) are the most efficient proof systems in terms of proof size and verification. Currently, Groth’s scheme from EUROCRYPT 2016, Groth16, is the state-of-the-art and is widely deployed in practice. Groth16 is originally proven to achieve knowledge soundness, which does not guarantee the non-malleability of proofs. There has been considerable progress in presenting new zk-SNARKs or modifying Groth16 to efficiently achieve strong Simulation extractability, which is shown to be a necessary requirement in some applications. In this paper, we revise the Random oracle based variant of Groth16 proposed by Bowe and Gabizon, BG18, the most efficient one in terms of prover efficiency and CRS size among the candidates, and present a more efficient variant that saves 2 pairings in the verification and 1 group element in the proof. This supersedes our preliminary construction, presented in CANS 2020 (Baghery et al. in CANS 20, volume 12579 of LNCS, Springer, Heidelberg. pp 453-461, 2020), which saved 1 pairing in the verification, and was proven in the generic group model. Our new construction also improves on BG18 in that our proofs are in the algebraic group model with Random Oracles and reduces security to standard computational assumptions in bilinear groups (as opposed to using the full power of the generic group model (GGM)). We implement our proposed simulation extractable zk-SNARK (SE zk-SNARK) along with BG18 in the Arkworks library, and compare the efficiency of our scheme with some related works. Our empirical experiences confirm that our SE zk-SNARK is more efficient than all previous simulation extractable (SE) schemes in most dimensions and it has very close efficiency to the original Groth16.

• Nunno, Giulia Di; Mishura, Yuliya & Yurchenko-Tytarenko, Anton (2023). Sandwiched SDEs with unbounded drift driven by Hölder noises. Advances in Applied Probability. ISSN 0001-8678. 55(3), s. 927–964. doi: 10.1017/apr.2022.56. Fulltekst i vitenarkiv Vis sammendrag

  We study a stochastic differential equation with an unbounded drift and general Hölder continuous noise of order . The corresponding equation turns out to have a unique solution that, depending on a particular shape of the drift, either stays above some continuous function or has continuous upper and lower bounds. Under some mild assumptions on the noise, we prove that the solution has moments of all orders. In addition, we provide its connection to the solution of some Skorokhod reflection problem. As an illustration of our results and motivation for applications, we also suggest two stochastic volatility models which we regard as generalizations of the CIR and CEV processes. We complete the study by providing a numerical scheme for the solution.

• Nunno, Giulia Di; Kubilius, Kestutis; Mishura, Yuliya & Yurchenko-Tytarenko, Anton (2023). From Constant to Rough: A Survey of Continuous Volatility Modeling. Mathematics. 11(19). doi: 10.3390/math11194201. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we present a comprehensive survey of continuous stochastic volatility models, discussing their historical development and the key stylized facts that have driven the field. Special attention is dedicated to fractional and rough methods: without advocating for either roughness or long memory, we outline the motivation behind them and characterize some landmark models. In addition, we briefly touch on the problem of VIX modeling and recent advances in the SPX-VIX joint calibration puzzle.

• Miao, Yingting; Rohde, Christian & Tang, Hao (2023). Well-posedness for a stochastic Camassa–Holm type equation with higher order nonlinearities. Stochastics and Partial Differential Equations: Analysis and Computations. ISSN 2194-0401. doi: 10.1007/s40072-023-00291-z. Fulltekst i vitenarkiv
• Tang, Hao & Yang, Anita (2023). Noise effects in some stochastic evolution equations: Global existence and dependence on initial data. Annales de l'I.H.P. Probabilites et statistiques. ISSN 0246-0203. 59(1), s. 378–410. doi: 10.1214/21-AIHP1241. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we consider the noise effects on a class of stochastic evolution equations including the stochastic Camassa–Holm equations with or without rotation. We first obtain the existence, uniqueness and a blow-up criterion of pathwise solutions in Sobolev space Hs with s>3/2 . Then we prove that strong enough noise can prevent blow-up with probability 1, which justifies the regularization effect of strong nonlinear noise in preventing singularities. Besides, such strengths of noise are estimated in different examples. Finally, for the interplay between regularization effect induced by the noise and the dependence on initial conditions, we introduce and investigate the stability of the exiting time and construct an example to show that the multiplicative noise cannot improve both the stability of the exiting time and the continuity of the dependence on initial data simultaneously. Dans cet article, nous considérons les effets du bruit sur une classe d’équations d’évolution stochastiques y compris les équations stochastiques de Camassa–Holm avec ou sans rotation. Nous obtenons d’abord l’existence, l’unicité et un critère d’explosion de solutions pathwise dans l’espace de Sobolev Hs avec s>3/2. Ensuite, nous prouvons qu’un bruit suffisamment fort peut empêcher l’explosion avec probabilité 1, ce qui justifie l’effet régularisant du bruit non linéaire fort dans la prévention des singularités. De plus, de telles forces de bruit sont estimées dans les différents exemples. Enfin, pour l’interaction entre l’effet de régularisation induit par le bruit et la dépendance par rapport aux conditions initiales, nous introduisons et étudions la stabilité du temps de sortie et construisons un exemple pour montrer que le bruit multiplicatif ne peut pas améliorer simultanément la stabilité du temps de sortie et la continuité de la dépendance par rapport aux données initiales.

• Harang, Fabian Andsem; Nilssen, Torstein & Proske, Frank Norbert (2022). GIRSANOV THEOREM FOR MULTIFRACTIONAL BROWNIAN PROCESSES. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. 94(8). doi: 10.1080/17442508.2022.2027948. Fulltekst i vitenarkiv Vis sammendrag

  In this article, we will present a new perspective on the variable-order fractional calculus, which allows for differentiation and integration to a variable order. The concept of multifractional calculus has been a scarcely studied topic within the field of functional analysis in the past 20 years. We develop a multifractional differential operator which acts as the inverse of the multifractional integral operator. This is done by solving the Abel integral equation generalized to a multifractional order. With this new multifractional differential operator, we prove a Girsanov's theorem for multifractional Brownian motions of Riemann–Liouville type. As an application, we show how Girsanov's theorem can then be applied to prove the existence of a unique strong solution to stochastic differential equations where the drift coefficient is merely of linear growth, and the driving noise is given by a non-stationary multifractional Brownian motion with a Hurst parameter as a function of time. The Hurst functions we study will take values in a bounded subset of (0,1/2) . The application of multifractional calculus to SDEs is based on a generalization of the works of D. Nualart and Y. Ouknine [Regularization of differential equations by fractional noise, Stoch Process Appl. 102(1) (2002), pp. 103–116].

• Nunno, Giulia Di (2022). On stochastic control for time changed Lévy dynamics. SeMA Journal - Boletin de la Sociedad Española de Matemática Aplicada. ISSN 2254-3902. 79, s. 529–547. doi: 10.1007/s40324-022-00301-5. Fulltekst i vitenarkiv Vis sammendrag

  Controlled stochastic differential equations driven by time changed Lévy noises do not enjoy the Markov property in general, but can be treated in the framework of general martingales. From the modelling point of view, time changed noises constitute a feasible way to include time dependencies at noise level and still keep a reasonably simple structure. Furthermore, they are easy to simulate, with the result that time change Lévy dynamics attract attention in various fields of application. In this work we survey an approach to stochastic control via maximum principle for time changed Lévy dynamics. We emphasise the role and use of different information flows in tackling the various control problems. We show how these techniques can be extended to include Volterra type dynamics and the control of forward–backward systems of equations. Our techniques make use of the stochastic non-anticipating (NA) derivative in a general martingale framework.

• Craig, Michael T.; Wohland, Jan; Stoop, Laurens P.; Kies, Alexander; Pickering, Bryn & Bloomfield, Hannah C. [Vis alle 23 forfattere av denne artikkelen] (2022). Overcoming the disconnect between energy system and climate modeling. Joule. 6(7), s. 1405–1417. doi: 10.1016/j.joule.2022.05.010. Fulltekst i vitenarkiv Vis sammendrag

  Energy system models underpin decisions by energy system planners and operators. Energy system modeling faces a transformation: accounting for changing meteorological conditions imposed by climate change. To enable that transformation, a community of practice in energy-climate modeling has started to form that aims to better integrate energy system models with weather and climate models. Here, we evaluate the disconnects between the energy system and climate modeling communities, then lay out a research agenda to bridge those disconnects. In the near-term, we propose interdisciplinary activities for expediting uptake of future climate data in energy system modeling. In the long-term, we propose a transdisciplinary approach to enable development of (1) energy-system-tailored climate datasets for historical and future meteorological conditions and (2) energy system models that can effectively leverage those datasets. This agenda increases the odds of meeting ambitious climate mitigation goals by systematically capturing and mitigating climate risk in energy sector decision-making.

• Benth, Fred Espen; Detering, Nils & Galimberti, Luca (2022). Neural networks in Fréchet spaces. Annals of Mathematics and Artificial Intelligence. ISSN 1012-2443. 91, s. 75–103. doi: 10.1007/s10472-022-09824-z. Fulltekst i vitenarkiv Vis sammendrag

  We propose a neural network architecture in infinite dimensional spaces for which we can show the universal approximation property. Indeed, we derive approximation results for continuous functions from a Fréchet space X into a Banach space Y. The approximation results are generalising the well known universal approximation theorem for continuous functions from Rn to R, where approximation is done with (multilayer) neural networks Cybenko (1989) Math. Cont. Signals Syst. 2, 303–314 and Hornik et al. (1989) Neural Netw., 2, 359–366 and Funahashi (1989) Neural Netw., 2, 183–192 and Leshno (1993) Neural Netw., 6, 861–867. Our infinite dimensional networks are constructed using activation functions being nonlinear operators and affine transforms. Several examples are given of such activation functions. We show furthermore that our neural networks on infinite dimensional spaces can be projected down to finite dimensional subspaces with any desirable accuracy, thus obtaining approximating networks that are easy to implement and allow for fast computation and fitting. The resulting neural network architecture is therefore applicable for prediction tasks based on functional data.

• Midtfjord, Alise Danielle; Bin, Riccardo De & Huseby, Arne Bang (2022). A boosting model for survival analysis with dependent censoring. I Torelli, Nicola; BELLIO, RUGGERO & MUGGEO, VITO (Red.), Proceedings of the 36th International Workshop on Statistical Modelling. EUT Edizioni Università di Trieste. ISSN 9788855113090. Fulltekst i vitenarkiv
• Dordevic, Jasmina (2022). A stochastic model for malaria and its behavior under insecticide-treated nets. Studies in applied mathematics (Cambridge). ISSN 0022-2526. 149(3), s. 631–656. doi: 10.1111/sapm.12515. Fulltekst i vitenarkiv Vis sammendrag

  Stochastic model for malaria transmission is introduced, and its behavior under insecticide-treated nets (which is used as a prevention) is considered. A white noise is introduced into the model, representing fluctuations in the environment that manifest themselves naturally on the transmission coefficient rate. Existence and uniqueness of a global positive solution of the stochastic model are proved, as well as the conditions under which extinction and persistence in mean hold are establish. Numerical simulations are provided, which illustrate that the theoretical results and conclusions are derived on the impact of the fluctuations that are caused by the environment.

• Schrader, Simon Elias & Benth, Fred Espen (2022). A stochastic study of carbon emission reduction from electrification and interconnecting cable utilization. The Norway and Germany case. Energy Economics. ISSN 0140-9883. 114. doi: 10.1016/j.eneco.2022.106300. Fulltekst i vitenarkiv Vis sammendrag

  We analyse how carbon emissions are reduced utilizing the new NordLink cable between Norway and Germany, taking policy plans of platform electrification in Norway into account. Building a stochastic model based on existing data for Norwegian and German electricity production and demand (Open Power system data, 2020) to account for uncertainty, we found that the cable can be used as an effective way to reduce German emissions by exporting German renewable surplus production on windy days to Norway and importing from Norway other days. We found that such flow of power will lead to a possible reduction of the countries’ total emissions by over 4.1 million tons CO annually. Based on a mean reduction of German CO emissions of about 710 g/kWh by electricity imported from Norway, we found that electrification of the Norwegian oil and gas extraction is inferior to an increased export to Germany in terms of total emission reduction, as electrification only gives a reduction of about 425 g/kWh according to numbers from the Norwegian Oil Directorate (Oljedirektoratet, 2020). A similar calculation assuming a realistic full electrification of the platforms only gave a reduction of 2.4 million tons CO annually. In the long term, we conclude that combining both approaches makes the best use of Norwegian surplus energy production in view of carbon emission reduction.

• Kovács, Mihály; Lang, Annika & Petersson, Andreas Erik (2022). Approximation of SPDE covariance operators by finite elements: a semigroup approach. IMA Journal of Numerical Analysis. ISSN 0272-4979. 43(3), s. 1324–1357. doi: 10.1093/imanum/drac020. Fulltekst i vitenarkiv Vis sammendrag

  Abstract The problem of approximating the covariance operator of the mild solution to a linear stochastic partial differential equation is considered. An integral equation involving the semigroup of the mild solution is derived and a general error decomposition is proven. This formula is applied to approximations of the covariance operator of a stochastic advection-diffusion equation and a stochastic wave equation, both on bounded domains. The approximations are based on finite element discretizations in space and rational approximations of the exponential function in time. Convergence rates are derived in the trace class and Hilbert–Schmidt norms with numerical simulations illustrating the results.

• Benth, Fred Espen & Galimberti, Luca (2022). Stochastic integrals and Gelfand integration in Fréchet spaces. Infinite Dimensional Analysis Quantum Probability and Related Topics. ISSN 0219-0257. 25(2), s. 1–35. doi: 10.1142/S0219025722500072. Fulltekst i vitenarkiv Vis sammendrag

  We provide a detailed analysis of the Gelfand integral on Fréchet spaces, showing among other things a Vitali theorem, dominated convergence and a Fubini result. Furthermore, the Gelfand integral commutes with linear operators. The Skorohod integral is conveniently expressed in terms of a Gelfand integral on Hida distribution space, which forms our prime motivation and example. We extend several results of Skorohod integrals to a general class of pathwise Gelfand integrals. For example, we provide generalizations of the Hida–Malliavin derivative and extend the integration-by-parts formula in Malliavin Calculus. A Fubini-result is also shown, based on the commutative property of Gelfand integrals with linear operators. Finally, our studies give the motivation for two existing definitions of stochastic Volterra integration in Hida space.

• Midtfjord, Alise Danielle; Bin, Riccardo De & Huseby, Arne Bang (2022). A decision support system for safer airplane landings: Predicting runway conditions using XGBoost and explainable AI. Cold Regions Science and Technology. ISSN 0165-232X. 199. doi: 10.1016/j.coldregions.2022.103556. Fulltekst i vitenarkiv Vis sammendrag

  The presence of snow and ice on runway surfaces reduces the available tire-pavement friction needed for retardation and directional control and causes potential economic and safety threats for the aviation industry during the winter seasons. To activate appropriate safety procedures, pilots need accurate and timely information on the actual runway surface conditions. In this study, XGBoost is used to create a combined runway assessment system, which includes a classification model to identify slippery conditions and a regression model to predict the level of slipperiness. The models are trained on weather data and runway reports. The runway surface conditions are represented by the tire-pavement friction coefficient, which is estimated from flight sensor data from landing aircrafts. The XGBoost models are combined with SHAP approximations to provide a reliable decision support system for airport operators, which can contribute to safer and more economic operations of airport runways. To evaluate the performance of the prediction models, they are compared to several state-of-the-art runway assessment methods. The XGBoost models identify slippery runway conditions with a ROC AUC of 0.95, predict the friction coefficient with a MAE of 0.0254, and outperforms all the previous methods. The results show the strong abilities of machine learning methods to model complex, physical phenomena with a good accuracy.

• Crisan, Dan; Lobbe, Alexander & Ortiz-Latorre, Salvador (2022). An application of the splitting-up method for the computation of a neural network representation for the solution for the filtering equations. Stochastics and Partial Differential Equations: Analysis and Computations. ISSN 2194-0401. 10, s. 1050–1081. doi: 10.1007/s40072-022-00260-y. Fulltekst i vitenarkiv Vis sammendrag

  Abstract The filtering equations govern the evolution of the conditional distribution of a signal process given partial, and possibly noisy, observations arriving sequentially in time. Their numerical approximation plays a central role in many real-life applications, including numerical weather prediction [Llopis et al. (SIAM J Sci Comput 40(3):A1544–A1565, 2018), Galanis et al. (Geophysicae 24(10): 2451–2460, 2006)], finance [Brigo and Hanzon (Insurance Math Econom 22(1):53–64, 1998), Date and Ponomareva (IMA J Manag Math 22(3): 195–211, 2011), Crisan and Rozovskii (The Oxford handbook of nonlinear filtering, 2011)] and engineering [Myötyri et al. (Reliability Eng Syst Saf 91(2):200–208, 2005)]. One of the classical approaches to approximate the solution of the filtering equations is to use a PDE inspired method, called the splitting-up method, initiated by Gyongy, Krylov, LeGland, among other contributors, see e.g., Gyöngy and Krylov (Stochastic inequalities and applications, Progr. Probab. 56:301–321, 2003), Le Gland(Stochastic partial differential equations and their applications (Charlotte, NC, 1991), Lect. Notes Control Inf. Sci. 176:177–187, 1992). This method, and other PDE based approaches, have particular applicability for solving low-dimensional problems. In this work we combine this method with a neural network representation inspired by [Han et al. (Proc Natl acad Sci 115(34):8505–8510, 2018)]. The new methodology is used to produce an approximation of the unnormalised conditional distribution of the signal process. We further develop a recursive normalisation procedure to recover the normalised conditional distribution of the signal process. The new scheme can be iterated over multiple time steps whilst keeping its asymptotic unbiasedness property intact. We test the neural network approximations with numerical approximation results for the Kalman and Benes filter.

• Eggen, Mari Dahl; Dahl, Kristina Rognlien; Näsholm, Sven Peter & Mæland, Steffen (2022). Stochastic Modeling of Stratospheric Temperature. Mathematical Geosciences. ISSN 1874-8961. 54, s. 651–678. doi: 10.1007/s11004-021-09990-6. Fulltekst i vitenarkiv Vis sammendrag

  This study suggests a stochastic model for time series of daily zonal (circumpolar) mean stratospheric temperature at a given pressure level. It can be seen as an extension of previous studies which have developed stochastic models for surface temperatures. The proposed model is a combination of a deterministic seasonality function and a Lévy-driven multidimensional Ornstein–Uhlenbeck process, which is a mean-reverting stochastic process. More specifically, the deseasonalized temperature model is an order 4 continuous-time autoregressive model, meaning that the stratospheric temperature is modeled to be directly dependent on the temperature over four preceding days, while the model’s longer-range memory stems from its recursive nature. This study is based on temperature data from the European Centre for Medium-Range Weather Forecasts ERA-Interim reanalysis model product. The residuals of the autoregressive model are well represented by normal inverse Gaussian-distributed random variables scaled with a time-dependent volatility function. A monthly variability in speed of mean reversion of stratospheric temperature is found, hence suggesting a generalization of the fourth-order continuous-time autoregressive model. A stochastic stratospheric temperature model, as proposed in this paper, can be used in geophysical analyses to improve the understanding of stratospheric dynamics. In particular, such characterizations of stratospheric temperature may be a step towards greater insight in modeling and prediction of large-scale middle atmospheric events, such as sudden stratospheric warming. Through stratosphere–troposphere coupling, the stratosphere is hence a source of extended tropospheric predictability at weekly to monthly timescales, which is of great importance in several societal and industry sectors.

• Huseby, Arne Bang (2022). Optimizing Multiple Reinsurance Contracts. I Leva, Maria Chiara; Patelli, Edoardo; Podofillini, Luca & Wilson, Simon (Red.), Proceedings of the 32nd European Safety and Reliability Conference (ESREL 2022). Research Publishing Services. ISSN 9789811851834. Fulltekst i vitenarkiv Vis sammendrag

  An insurance contract implies that risk is ceded from ordinary policy holders to companies. Companies do the same thing between themselves, and this is known as reinsurance. The problem of determining reinsurance contracts which are optimal with respect to some reasonable criterion has been studied extensively. Different contract types are considered such as stop-loss contracts where the reinsurance company covers risk above a certain level, and insurance layer contracts where the reinsurance company covers risk within an interval. The contracts are then optimized with respect to some risk measure, such as value-at-risk or conditional tail expectation. In the present paper we investigate this problem further and show that the optimal solution depends on the tail hazard rates of the risk distributions. If the tail hazard rates are decreasing, which is the case for heavy tailed distributions like lognormal and pareto distributions, the optimal solution is balanced. That is, reinsurance contracts for identically distributed risks should be identical insurance layer contracts. However, if the tail hazard rate is increasing, which is the case for light tailed distributions like truncated normal distributions, the optimal solution is typically not balanced. Even for identically distributed risks, some contracts should be insurance layer contracts, while others should be stop-loss contracts. In the limiting case, where the hazard rate is constant, i.e., when the risks are exponentially distributed, we show that a balanced solution is optimal. We also present an efficient importance sampling method for estimating optimal contracts.

• Galeati, Lucio & Harang, Fabian Andsem (2022). Regularization of multiplicative SDEs through additive noise. The Annals of Applied Probability. ISSN 1050-5164. 32(5), s. 3930–3963. doi: 10.1214/21-AAP1778. Fulltekst i vitenarkiv Vis sammendrag

  We investigate the regularizing effect of certain additive continuous perturbations on SDEs with multiplicative fractional Brownian motion (fBm). Traditionally, a Lipschitz requirement on the drift and diffusion coefficients is imposed to ensure existence and uniqueness of the SDE. We show that suitable perturbations restore existence, uniqueness and regularity of the flow for the resulting equation, even when both the drift and the diffusion coefficients are distributional, thus extending the program of regularization by noise to the case of multiplicative SDEs. Our method relies on a combination of the nonlinear Young formalism developed by Catellier and Gubinelli (Stochastic Process. Appl. 126 (2016) 2323–2366), and stochastic averaging estimates recently obtained by Hairer and Li (Ann. Probab. 48 (2020) 1826–1860).

• Dahl, Kristina Rognlien; Huseby, Arne Bang & Havgar, Marius (2022). Optimal Reinsurance Contracts under Conditional Value-at-risk. I Leva, Maria Chiara; Patelli, Edoardo; Podofillini, Luca & Wilson, Simon (Red.), Proceedings of the 32nd European Safety and Reliability Conference (ESREL 2022). Research Publishing Services. ISSN 9789811851834. Fulltekst i vitenarkiv Vis sammendrag

  An insurance contract implies that risk is ceded from ordinary policy holders to companies. However, companies do the same thing between themselves, and this is known as reinsurance. The problem of determining reinsurance contracts which are optimal with respect to some reasonable criterion has been studied extensively within actuarial science. Different contract types are considered such as stop-loss contracts where the reinsurance company covers risk above a certain level, and insurance layer contracts where the reinsurance company covers risk within an interval. The contracts are then optimized with respect to some risk measure, such as value-at-risk or conditional value-at- risk. In the present paper we consider the problem of minimizing conditional value-at-risk in the case of multiple stop-loss contracts. Such contracts are known to be optimal in the univariate case, and the optimal contract is easily determined. We show that the same holds in the multivariate case, both with dependent and independent risks. The results are illustrated with some numerical examples.

• Kovács, Mihály; Lang, Annika & Petersson, Andreas Erik (2022). Hilbert–Schmidt regularity of symmetric integral operators on bounded domains with applications to SPDE approximations. Stochastic Analysis and Applications. ISSN 0736-2994. doi: 10.1080/07362994.2022.2053541. Fulltekst i vitenarkiv Vis sammendrag

  Regularity estimates for an integral operator with a symmetric continuous kernel on a convex bounded domain are derived. The covariance of a mean-square continuous random field on the domain is an example of such an operator. The estimates are of the form of Hilbert–Schmidt norms of the integral operator and its square root, composed with fractional powers of an elliptic operator equipped with homogeneous boundary conditions of either Dirichlet or Neumann type. These types of estimates, which couple the regularity of the driving noise with the properties of the differential operator, have important implications for stochastic partial differential equations on bounded domains as well as their numerical approximations. The main tools used to derive the estimates are properties of reproducing kernel Hilbert spaces of functions on bounded domains along with Hilbert–Schmidt embeddings of Sobolev spaces. Both non-homogeneous and homogeneous kernels are considered. In the latter case, results in a general Schatten class norm are also provided. Important examples of homogeneous kernels covered by the results of the paper include the class of Matérn kernels.

• Alonso-Orán, Diego; Miao, Yingting & Tang, Hao (2022). Global existence, blow-up and stability for a stochastic transport equation with non-local velocity. Journal of Differential Equations. ISSN 0022-0396. 335, s. 244–293. doi: 10.1016/j.jde.2022.06.025. Fulltekst i vitenarkiv Vis sammendrag

  In this paper we investigate a non-linear and non-local one dimensional transport equation under random perturbations on the real line. We first establish a local-in-time theory, i.e., existence, uniqueness and blow-up criterion for pathwise solutions in Sobolev spaces Hs with s>3. Thereafter, we give a picture of the long time behavior of the solutions based on the type of noise we consider. On one hand, we identify a family of noises such that blow-up can be prevented with probability 1, guaranteeing the existence and uniqueness of global solutions almost surely. On the other hand, in the particular linear noise case, we show that singularities occur in finite time with positive probability, and we derive lower bounds of these probabilities. To conclude, we introduce the notion of stability of exiting times and show that one cannot improve the stability of the exiting time and simultaneously improve the continuity of the dependence on initial data.

• Dordevic, Jasmina & Dahl, Kristina Rognlien (2022). Stochastic optimal control of pre-exposure prophylaxis for HIV infection. Mathematical Medicine and Biology. ISSN 1477-8599. 39(3), s. 197–225. doi: 10.1093/imammb/dqac003. Fulltekst i vitenarkiv
• Orvieto, Antonio; Kersting, Hans; Proske, Frank Norbert; Bach, Francis & Lucchi, Aurelien (2022). Anticorrelated Noise Injection for Improved Generalization. Proceedings of Machine Learning Research (PMLR). 162, s. 17094–17116. Fulltekst i vitenarkiv Vis sammendrag

  Injecting artificial noise into gradient descent (GD) is commonly employed to improve the performance of machine learning models. Usually, uncorrelated noise is used in such perturbed gradient descent (PGD) methods. It is, however, not known if this is optimal or whether other types of noise could provide better generalization performance. In this paper, we zoom in on the problem of correlating the perturbations of consecutive PGD steps. We consider a variety of objective functions for which we find that GD with anticorrelated perturbations ("Anti-PGD") generalizes significantly better than GD and standard (uncorrelated) PGD. To support these experimental findings, we also derive a theoretical analysis that demonstrates that Anti-PGD moves to wider minima, while GD and PGD remain stuck in suboptimal regions or even diverge. This new connection between anticorrelated noise and generalization opens the field to novel ways to exploit noise for training machine learning models.

• Coffie, Emmanuel; Mao, Xuerong & Proske, Frank Norbert (2022). On the Analysis of a Generalised Rough Ait-Sahalia Interest Rate Model. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv
• Mishura, Yuliya & Yurchenko-Tytarenko, Anton (2022). Standard and fractional reflected Ornstein–Uhlenbeck processes as the limits of square roots of Cox–Ingersoll–Ross processes. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. doi: 10.1080/17442508.2022.2047188. Fulltekst i vitenarkiv
• Benth, Fred Espen; Schroers, Dennis & Veraart, Almut E. D. (2022). A weak law of large numbers for realised covariation in a Hilbert space setting. Stochastic Processes and their Applications. ISSN 0304-4149. 145, s. 241–268. doi: 10.1016/j.spa.2021.12.011. Fulltekst i vitenarkiv Vis sammendrag

  This article generalises the concept of realised covariation to Hilbert-space-valued stochastic processes. More precisely, based on high-frequency functional data, we construct an estimator of the trace-class operator-valued integrated volatility process arising in general mild solutions of Hilbert space-valued stochastic evolution equations in the sense of Da Prato and Zabczyk (2014). We prove a weak law of large numbers for this estimator, where the convergence is uniform on compacts in probability with respect to the Hilbert–Schmidt norm. In addition, we determine convergence rates for common stochastic volatility models in Hilbert spaces.

• Agram, Nacira; Labed, Saloua; Øksendal, Bernt Karsten & Yakhlef, Samia (2022). Singular Control of Stochastic Volterra Integral Equations. Acta Mathematica Scientia. ISSN 0252-9602. 42(3), s. 1003–1017. doi: 10.1007/s10473-022-0311-9. Fulltekst i vitenarkiv
• Benth, Fred Espen; Nunno, Giulia Di & Schroers, Dennis (2022). A topological proof of Sklar's theorem in arbitrary dimensions. Dependence Modeling. 10(1), s. 22–28. doi: 10.1515/demo-2022-0103. Fulltekst i vitenarkiv Vis sammendrag

  Abstract Copulas are appealing tools in multivariate probability theory and statistics. Nevertheless, the transfer of this concept to infinite dimensions entails some nontrivial topological and functional analytic issues, making a deeper theoretical understanding indispensable toward applications. In this short work, we transfer the well-known property of compactness of the set of copulas in finite dimensions to the infinite-dimensional framework. As an application, we prove Sklar’s theorem in infinite dimensions via a topological argument and the notion of inverse systems.

• Coffie, Emmanuel; Duedahl, Sindre & Proske, Frank Norbert (2022). Sensitivity analysis with respect to a stochastic stock price model with rough volatility via a Bismut-Elworthy-Li formula for singular SDEs. Stochastic Processes and their Applications. ISSN 0304-4149. 156, s. 156–195. doi: 10.1016/j.spa.2022.11.001. Fulltekst i vitenarkiv
• Benth, Fred Espen; Detering, Nils & Krühner, Paul (2022). Stochastic Volterra integral equations and a class of first-order stochastic partial differential equations. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. s. 1–23. doi: 10.1080/17442508.2021.2019738. Fulltekst i vitenarkiv Vis sammendrag

  We investigate stochastic Volterra equations and their limiting laws. The stochastic Volterra equations we consider are driven by a Hilbert space valued Lévy noise and integration kernels may have non-linear dependence on the current state of the process. Our method is based on an embedding into a Hilbert space of functions which allows to represent the solution of the Volterra equation as the boundary value of a solution to a stochastic partial differential equation. We first gather abstract results and give more detailed conditions in more specific function spaces.

• Lucchi, Aurelien; Proske, Frank Norbert; Orvieto, Antonio; Bach, Francis & Kersting, Hans (2022). On the Theoretical Properties of Noise Correlation in Stochastic Optimization. Advances in Neural Information Processing Systems. ISSN 1049-5258. Fulltekst i vitenarkiv Vis sammendrag

  Studying the properties of stochastic noise to optimize complex non-convex functions has been an active area of research in the field of machine learning. Prior work~\citep{zhou2019pgd, wei2019noise} has shown that the noise of stochastic gradient descent improves optimization by overcoming undesirable obstacles in the landscape. Moreover, injecting artificial Gaussian noise has become a popular idea to quickly escape saddle points. Indeed, in the absence of reliable gradient information, the noise is used to explore the landscape, but it is unclear what type of noise is optimal in terms of exploration ability. In order to narrow this gap in our knowledge, we study a general type of continuous-time non-Markovian process, based on fractional Brownian motion, that allows for the increments of the process to be correlated. This generalizes processes based on Brownian motion, such as the Ornstein-Uhlenbeck process. We demonstrate how to discretize such processes which gives rise to the new algorithm ``fPGD''. This method is a generalization of the known algorithms PGD and Anti-PGD~\citep{orvieto2022anti}. We study the properties of fPGD both theoretically and empirically, demonstrating that it possesses exploration abilities that, in some cases, are favorable over PGD and Anti-PGD. These results open the field to novel ways to exploit noise for training machine learning models.

• Crisan, Dan; Lobbe, Alexander & Ortiz-Latorre, Salvador (2022). Pathwise Approximations for the Solution of the Non-Linear Filtering Problem. I Yin, George & Zariphopoulou, Thaleia (Red.), Stochastic Analysis, Filtering, and Stochastic Optimization - A Commemorative Volume to Honor Mark H. A. Davis’s Contributions. Springer Nature. ISSN 9783030985189. s. 79–99. doi: 10.1007/978-3-030-98519-6_4. Fulltekst i vitenarkiv
• Savku, Emel (2022). Fundamentals of Market Making Via Stochastic Optimal Control. I Purutçuoğlu, Vilda; Weber, Gerhard-Wilhelm & Farnoudkia, Hajar (Red.), Operations Research: New Paradigms and Emerging Applications. CRC Press. ISSN 9781032349268. s. 136–154. Fulltekst i vitenarkiv
• Galeati, Lucio; Harang, Fabian Andsem & Mayorcas, Avi (2022). Distribution dependent SDEs driven by additive continuous noise. Electronic Journal of Probability (EJP). 27, s. 1–38. doi: 10.1214/22-EJP756. Fulltekst i vitenarkiv Vis sammendrag

  We study distribution dependent stochastic differential equation driven by a continuous process, without any specification on its law, following the approach initiated in [17]. We provide several criteria for existence and uniqueness of solutions which go beyond the classical globally Lipschitz setting. In particular we show well-posedness of the equation, as well as almost sure convergence of the associated particle system, for drifts satisfying either Osgood-continuity, monotonicity, local Lipschitz or Sobolev differentiability type assumptions.

• Dahl, Kristina Rognlien & Eyjolfsson, Heidar (2022). Self-exciting jump processes and their asymptotic behaviour. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. doi: 10.1080/17442508.2022.2028789. Fulltekst i vitenarkiv Vis sammendrag

  The purpose of this paper is to investigate properties of self-exciting jump processes where the intensity is given by an SDE, which is driven by a finite variation stochastic jump process. The value of the intensity process immediately before a jump may influence the jump size distribution. We focus on properties of this intensity function, and show that for each fixed point in time, t≥0, a scaling limit of the intensity process converges in distribution, and the limit equals the strong solution of the square-root diffusion process (Cox–Ingersoll–Ross process) at t. As a particular example, we study the case of a linear intensity process and derive explicit expressions for the expectation and variance in this case.

• Nunno, Giulia Di; Mishura, Yuliya & Yurchenko-Tytarenko, Anton (2022). Drift-implicit Euler scheme for sandwiched processes driven by Hölder noises. Numerical Algorithms. ISSN 1017-1398. doi: 10.1007/s11075-022-01424-6. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we analyze the drift-implicit (or backward) Euler numerical scheme for a class of stochastic differential equations with unbounded drift driven by an arbitrary λ-Hölder continuous process, λ ∈ (0,1). We prove that, under some mild moment assumptions on the Hölder constant of the noise, the Lr(Ω;L∞([0,T]))-approximation error converges to 0 as O(Δλ), Δ → 0. To exemplify, we consider numerical schemes for the generalized Cox–Ingersoll–Ross and Tsallis–Stariolo–Borland models. The results are illustrated by simulations.

• Galeati, Lucio; Harang, Fabian Andsem & Mayorcas, Avi (2022). Distribution dependent SDEs driven by additive fractional Brownian motion. Probability theory and related fields. ISSN 0178-8051. s. 1–59. doi: 10.1007/s00440-022-01145-w. Fulltekst i vitenarkiv
• Harang, Fabian Andsem; Tindel, Samy & Wang, Xiaohua (2022). Volterra equations driven by rough signals 2: higher order expansions. Stochastics and Dynamics. ISSN 0219-4937. doi: 10.1142/S0219493723500028. Fulltekst i vitenarkiv Vis sammendrag

  We extend the recently developed rough path theory for Volterra equations from [12] to the case of more rough noise and/or more singular Volterra kernels. It was already observed in [12] that the Volterra rough path introduced there did not satisfy any geometric relation, similar to that observed in classical rough path theory. Thus, an extension of the theory to more irregular driving signals requires a deeper understanding of the specific algebraic structure arising in the Volterra rough path. Inspired by the elements of \non-geometric rough paths" developed in [11] and [14] we provide a simple description of the Volterra rough path and the controlled Volterra process in terms of rooted trees, and with this description we are able to solve rough Volterra equations driven by more irregular signals.

• Agram, Nacira; Hu, Yaozhong & Øksendal, Bernt Karsten (2022). Mean-field backward stochastic differential equations and applications. Systems & control letters (Print). ISSN 0167-6911. 162. doi: 10.1016/j.sysconle.2022.105196. Fulltekst i vitenarkiv
• Pilipenko, Andrey & Proske, Frank Norbert (2021). Small Noise Perturbations in Multidimensional Case. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv
• Coffie, Emmanuel; Duedahl, Sindre & Proske, Frank Norbert (2021). Sensitivity Analysis with respect to a Stock Price Model with Rough Volatility via a Bismut-Elworthy-Li Formula for Singular SDEs. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv
• Coffie, Emmanuel; Duedahl, Sindre & Proske, Frank Norbert (2021). Thiele's Differential Equation Based on Markov Jump Processes with Non-countable State Space. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv
• Harang, Fabian Andsem & Catellier, Rémi (2021). Pathwise regularization of the stochastic heat equation with multiplicative noise through irregular perturbation. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv
• Midtfjord, Alise Danielle; Bin, Riccardo De & Huseby, Arne Bang (2021). A Machine Learning Approach to Safer Airplane Landings: Predicting Runway Conditions using Weather and Flight Data. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv Vis sammendrag

  The presence of snow and ice on runway surfaces reduces the available tire-pavement friction needed for retardation and directional control and causes potential economic and safety threats for the aviation industry. To activate appropriate safety procedures, pilots need accurate and timely information on the actual runway surface conditions. In this study, XGBoost is used to create a combined runway assessment system, which includes a classification model to identify slippery conditions and a regression model to predict the level of slipperiness. The models are trained on weather data and runway reports. The runway surface conditions are represented by the tire-pavement friction coefficient, which is estimated from flight sensor data from landing aircrafts. The XGBoost models are combined with SHAP approximations to provide a reliable decision support system for airport operators and pilots, which can contribute to safer and more economic operations of airport runways. To evaluate the performance of the prediction models, they are compared to several state-of-the-art runway assessment methods. The XGBoost models identify slippery runway conditions with a ROC AUC of 0.95, predict the friction coefficient with a MAE of 0.0254, and outperforms all the previous methods. The results show the strong abilities of machine learning methods to model complex, physical phenomena with a good accuracy.

• Crisan, Dan; Kurtz, Thomas G. & Ortiz-Latorre, Salvador (2021). Particle Representation for the Solution of the Filtering Problem. Application to the Error Expansion of Filtering Discretizations. Journal of Stochastic Analysis (JOSA). doi: 10.31390/josa.2.3.15. Fulltekst i vitenarkiv Vis sammendrag

  We introduce a weighted particle representation for the solution of the filtering problem based on a suitably chosen variation of the classical de Finetti theorem. This representation has important theoretical and numerical applications. In this paper, we explore some of its theoretical con- sequences. The first is to deduce the equations satisfied by the solution of the filtering problem in three different frameworks: the signal independent Brownian measurement noise model, the spatial observations with additive white noise model and the cluster detection model in spatial point processes. Secondly we use the representation to show that a suitably chosen filtering discretisation converges to the filtering solution. Thirdly we study the leading error coefficient for the discretisation. We show that it satisfies a stochastic partial differential equation by exploiting the weighted particle representation for both the approximation and the limiting filtering solution.

• Dahl, Kristina Rognlien & Eyjolfsson, Heidar (2021). Self-Exciting Jump Processes as Deterioration Models. I Castanier, Bruno; Cepin, Marko; Bigaud, David & Berenguer, Christophe (Red.), Proceedings of the 31st European Safety and Reliability Conference. Research Publishing Services. ISSN 9789811820168. doi: 10.3850/978-981-18-2016-8_286-cd. Fulltekst i vitenarkiv Vis sammendrag

  Several different approaches to modelling stochastic deterioration for optimising maintenance have been suggested in the reliability literature. These include component lifetime distributions, which have the disadvantage of being binary, in the sense of only telling whether the component has failed or not. Failure rate functions model ageing in a more satisfactory way than lifetime distributions. However, failure rates cannot be observed for a single component, and are therefore not tractable in practical applications. To mitigate this, a theory for modeling deterioration via stochastic processes developed. Various processes have been suggested, such as Brownian motion with drift and compound Poisson processes (CPP) for modeling usage and damage from sporadic shocks and gamma processes to model gradual ageing. However, none of these processes are able to capture jump clustering. To allow for clustering of jumps (failure events), we suggest an alternative approach in this paper: To use self-exciting jump processes to model stochastic deterioration of components in a system where there may be clustering effects in the degradation. Self-exciting processes excite their own intensity, so large shocks are likely to be followed by another shock within a short period of time. Furthermore, self-exciting processes may have both finite and infinite activity. Therefore, we suggest that these processes can be used to model degradation both by sporadic shocks and by gradual wear. We illustrate the use of self-exciting degradation with several numerical examples. In particular, we use Monte Carlo simulation to estimate the expected lifetime of a component with self-exciting degradation. As an illustration, we also estimate the lifetime of a bridge system with independent components with identically distributed self-exciting degradation.

• Benth, Fred Espen & Simonsen, Iben Cathrine (2021). Metatimes, random measures and cylindrical random variables. Modern Stochastics: Theory and Applications (MSTA). ISSN 2351-6046. 8(3), s. 349–371. doi: 10.15559/21-VMSTA178. Fulltekst i vitenarkiv Vis sammendrag

  Metatimes constitute an extension of time-change to general measurable spaces, defined as mappings between two σ-algebras. Equipping the image σ-algebra of a metatime with a measure and defining the composition measure given by the metatime on the domain σ-algebra, we identify metatimes with bounded linear operators between spaces of square integrable functions. We also analyse the possibility to define a metatime from a given bounded linear operator between Hilbert spaces, which we show is possible for invertible operators. Next we establish a link between orthogonal random measures and cylindrical random variables following a classical construction. This enables us to view metatime-changed orthogonal random measures as cylindrical random variables composed with linear operators, where the linear operators are induced by metatimes. In the paper we also provide several results on the basic properties of metatimes as well as some applications towards trawl processes.

• Harang, Fabian Andsem & Mayorcas, Avi (2020). Pathwise regularisation of singular interacting particle systems and their mean field limits. Stochastic Processes and their Applications. ISSN 0304-4149. 159, s. 499–540. doi: 10.1016/j.spa.2023.02.005. Fulltekst i vitenarkiv

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• Nunno, Giulia Di; Mishura, Yuliya & Yurchenko-Tytarenko, Anton (2026). Volterra Volatility Models: Option Pricing and Hedging. Springer Nature. ISBN 9783032265753. doi: 10.1007/978-3-032-26576-0. Fulltekst i vitenarkiv Vis sammendrag

  This monograph introduces flexible Hölder‑noise models capturing rough and long‑range dependent volatility, with practical pricing and hedging methods.

• Benth, Fred Espen & Krühner, Paul (2023). Stochastic Models for Prices Dynamics in Energy and Commodity Markets An Infinite-Dimensional Perspective. Springer. ISBN 9783031403668. 250 s. Fulltekst i vitenarkiv

Se alle arbeider i NVA

• Ortiz-Latorre, Salvador (2026). Heston-Hawkes stochastic volatility model: Change of measure and Thiele's PIDE. Fulltekst i vitenarkiv
• Ortiz-Latorre, Salvador (2026). Convergence in law for quasi-linear SPDEs. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2026). Uncertainties in risk evaluation for long term horizons and computational aspects. doi: https:/www.bacheliercongress.com/2026/eventi.unibo.it/bachelier.html. Fulltekst i vitenarkiv Vis sammendrag

  Long and short-horizons must be treated differently in monetary risk-evaluations. Horizon-risk and h-longevity are identified to capture this. We clarify that classical dynamic risk measures are subject to horizon-risk exposure, and propose to work with fully-dynamic risk-measures. In long horizons, also interest rate uncertainty cannot be underestimated. We combine these aspect working with cash non-additive dynamic risk-measures via BSDEs and generalised shortfalls. We take steps in the numerical quantification relying on a novel use of Wiener-Ito chaos expansions and proposing DeepO BSDEs schemes.

• Nunno, Giulia Di (2026). Optimal control of Volterra dynamics: from finite to infinite dimensions in UMD Banach spaces and return. doi: https:/sites.google.com/view/trr388/events/past-events/workshop-optimal-control-2026?authuser=1. Fulltekst i vitenarkiv Vis sammendrag

  We consider a stochastic optimal control problem for Volterra type forward dynamics, which are typically non-Markovian in nature. A direct application of dynamic pro- gramming is not feasible. So we propose to lift the problem to infinite dimensions, where we can recover Markovianity. For this we work with Banach-valued processes. In this framework we tackle the optimal control problem by studying the corresponding Hamilton-Jacobi-Bellman equation, which can be formulated thanks to the appropriate lift. Our techniques make use of Malliavin calculus in UMD Banach spaces. Finally, we reconnect the solution of the infinite dimensional problem to the original finite dimen- sional one.

• Nunno, Giulia Di (2026). BSDEs and fully-dynamic risk measures. doi: https:/fdnss.fi/44th-finnish-summer-school-on-probability-and-statistics-50-years-jubilee/. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2026). Malliavin calculus in Sandwich Volterra Volatility models. Fulltekst i vitenarkiv Vis sammendrag

  Sandwich Volterra Volatility models are capturing both the long-memory and rough features of volatility, while also complying with stylised facts observed in volatility markets. Within the analysis of this family, we see how Malliavin calculus has been instrumental to characterise the behaviour of these processes. As supplement, we also discuss generalised product and chain rule.

• Nunno, Giulia Di (2026). BSDEs and Operator Deep BSDEs for dynamic risk evaluation. doi: https:/www.ucc.ie/en/stochastica/events/stochasticadirectionsworkshopapril2026/. Fulltekst i vitenarkiv Vis sammendrag

  Backward SDEs (BSDEs) have been used for dynamic risk measurement, we revise the framework and value of a dynamic risk evaluation. We also indicate some shortcomings, particularly related to the evaluation of risk at different time horizon, introduced as horizon-risk. For this reason we introduce the concept of fully-dynamic risk measurement and show how it is still easily related to BSDEs. As illustration we consider variation to the entropic case as well as we introduce the hq-entropic risk measure inspired by Tsallis general entropy. We discuss computational methods from an operator point of view, proposing an Operator Deep BSDE approach.

• Nunno, Giulia Di (2026). Capturing horizon risk and interest rate uncertainty. doi: https:/sites.google.com/view/mf-osaka/home?authuser=1. Fulltekst i vitenarkiv Vis sammendrag

  Horizon risk is introduced as the evaluation of the exposure to potentially erroneous risk measurement when using a risk measure that is not adequate to the actual time horizon of the position. We clarify that classical dynamic risk measures are subject to horizon risk, so we propose to work with fully-dynamic risk measures, which are then designed to capture horizon risk and evaluate it in terms of an index of h-longevity. For this we use backward stochastic differential equations. In a financial context, we combine horizon risk with interest rates uncertainty, thus we also propose a cash non-additive version of fully- dynamic risk measures. We construct such risk measures via backward stochastic differential equations or via a generalisation of shortfall-type representation. As illustration, we introduce the class of hq-entropic risk measures.

• Gancia, Gino; Mittal, Shivika; Nunno, Giulia Di; Castellacci, Fulvio & Beferull-Lozano, Baltasar (2026). SURE-AI@STAR Workshop on the Environmental Cost of AI. doi: https:/www.mn.uio.no/math/english/research/groups/risk-stochastics/events/seminars/SURE-AI@STARWorkshop-CostAI.html. Fulltekst i vitenarkiv
• Tarquini, Leonardo (2026). Signature McKean-Vlasov stochastic differential equations. doi: https:/sites.google.com/unipv.it/apmp2026/home-page. Fulltekst i vitenarkiv Vis sammendrag

  McKean--Vlasov-type stochastic differential equations (SDEs) are characterised by coefficients depending on both the state and the law of the solution. In our work, we focus on a class of such equations where the coefficients depend on a linear combination of the expected signature of the geometric p-rough path lift of its solution, with p ∈ (2,3). After establishing the strong existence and uniqueness of a solution, we prove how such an equation can approximate a general class of path-dependent McKean–Vlasov SDEs. Finally, we consider the associated particle system and establish propagation of chaos. This is a joint work with F.E. Benth (BI Norwegian Business School and University of Oslo) and S. Ortiz-Latorre (University of Oslo).

• Lozano, Pere Diaz (2026). Deep operator BSDE: A numerical scheme to approximate solution operators. Fulltekst i vitenarkiv Vis sammendrag

  Motivated by dynamic risk measures and conditional g-expectations, in this work we propose a numerical method to approximate the solution operator given by a backward stochastic differential equation (BSDE). The main ingredients for this are the Wiener chaos decomposition and the classical Euler scheme for BSDEs. We show convergence of this scheme under very mild assumptions, and provide a rate of convergence in more restrictive cases. We then implement it using neural networks, and we present several numerical examples where we can check the accuracy of the method.

• Lozano, Pere Diaz (2026). A Wiener Chaos Approach to Martingale Modelling and Implied Volatility Calibration. Fulltekst i vitenarkiv Vis sammendrag

  Calibration to a surface of option prices requires specifying a suitably flexible martingale model for the discounted asset price under a risk-neutral measure. Assuming Brownian noise and mean-square integrability, we construct an over-parameterized model based on the martingale representation theorem. In particular, we approximate the terminal value of the martingale via a truncated Wiener--chaos expansion and recover the intermediate dynamics by computing the corresponding conditional expectations. Using the Hermite-polynomial formulation of the Wiener chaos, we obtain easily implementable expressions that enable fast calibration to a target implied-volatility surface. We illustrate the flexibility and expressive power of the resulting model through numerical experiments on both simulated and real market data.

• Lozano, Pere Diaz (2026). A Wiener Chaos Approach to Martingale Modelling and Implied Volatility Calibration. Fulltekst i vitenarkiv Vis sammendrag

  Calibration to a surface of option prices requires specifying a suitably flexible martingale model for the discounted asset price under a risk-neutral measure. Assuming Brownian noise and mean-square integrability, we construct an over-parameterized model based on the martingale representation theorem. In particular, we approximate the terminal value of the martingale via a truncated Wiener--chaos expansion and recover the intermediate dynamics by computing the corresponding conditional expectations. Using the Hermite-polynomial formulation of the Wiener chaos, we obtain easily implementable expressions that enable fast calibration to a target implied-volatility surface. We illustrate the flexibility and expressive power of the resulting model through numerical experiments on both simulated and real market data.

• Lozano, Pere Diaz (2026). An Euler scheme for BSDEs via the Wiener chaos decomposition. Fulltekst i vitenarkiv Vis sammendrag

  The Euler scheme is a standard time discretization for BSDEs, but its implementation hinges on approximating conditional expectations and the associated martingale terms at each time step. We propose an implementation based on the Wiener chaos decomposition to approximate these quantities. In contrast to many numerical schemes that rely on a finite-dimensional Markovian representation, our approach accommodates arbitrary FT-measurable square-integrable terminal conditions. We provide a comprehensive convergence analysis under additional Malliavin regularity assumptions and illustrate the method on several numerical examples, including genuinely non-Markovian problems arising, for instance, in the pricing and hedging of contingent claims under rough-volatility models.

• Nunno, Giulia Di (2026). On Sandwiched Volterra Volatility Models. Fulltekst i vitenarkiv Vis sammendrag

  Sandwiched Volterra Volatility (SVV) models are a class of dynamics capable of capturing both the long-memory and rough features of volatility, while also complying with several stylized facts observed in volatility markets. At the same time, price dynamics with SVV volatility remain sufficiently tractable to allow for a rigorous and explicit analysis of pricing and hedging of financial assets. We present an overview of these models, highlighting their main features and potential applications. While pricing aspects are briefly discussed, the main focus of the talk is on the quadratic hedging of financial derivatives within this framework, providing insights into the computational challenges associated with the optimal hedging strategy.

• Nunno, Giulia Di (2026). Quadratic hedging in Sandwiched Volterra Volatility Models. Fulltekst i vitenarkiv Vis sammendrag

  Sandwiched Volterra Volatility (SVV) models are a class of dynamics capable of capturing both the long-memory and rough features of volatility, while also complying with several stylized facts observed in volatility markets. At the same time, price dynamics with SVV volatility remain sufficiently tractable to allow for a rigorous and explicit analysis of pricing and hedging of financial assets. We present an overview of these models, highlighting their main features and potential applications. While pricing aspects are briefly discussed, the main focus of the talk is on the quadratic hedging of financial derivatives within this framework, providing insights into the computational challenges associated with the optimal hedging strategy. The explicit optimal hedging solution is characterized via the non-anticipating derivative, and its numerical implementation is addressed through Markovian approximations.

• Nunno, Giulia Di (2025). Stochastic control for time-changed Brownian dynamics via change of variable formulae. doi: https:/www.dm.uniba.it/it/ricerca/convegni/2025/contropt2025. Fulltekst i vitenarkiv Vis sammendrag

  We derive novel change of variable formulae for stochastic integrals w.r.t. a time-changed Brownian motion where we assume that the time-change is a general increasing stochastic process with finitely many jumps in a bounded set of the positive half-line and is independent of the Brownian motion. These results are crucial in the study of optimal stochastic control in a semimartingale setting, where the semimartingale is written in terms of a time-changed Brownian motion and a finite variation process. Our techiques include the use of an initial enlargement of filtrations. Optimal strategies are derived for power and logarithmic utilities.

• Nunno, Giulia Di (2025). Stochastic optimal control for quadratic hedging in presence of memory and roughness. doi: https:/events.unibo.it/irreg-stoch-indam-cortona-25. Fulltekst i vitenarkiv Vis sammendrag

  Sandwiched Volterra Volatility (SVV) models are a class of models able to capture both the long memory and the rough aspects of volatility as well as complying with many of the stylised features typical of volatilities. We present the model and its properties. We then discuss questions of quadratic hedging. Particularly, while the theoretical solution is characterised in terms of the non-anticipating derivative for all square integrable claims, the fact that these models are typically non-Markovian provides a concrete difficulty in the direct computation of conditional expectations at the core of the explicit hedging strategy. To overcome this difficulty, we propose a Markovian approximation of the model which stems from an adequate approximation of the kernel in the Volterra noise.

• Nunno, Giulia Di (2025). Capturing cash non-additivity and horizon risk. doi: https:/conference2025.streamlit.app/. Fulltekst i vitenarkiv Vis sammendrag

  Horizon risk is the use of risk measures designed for short term position to evaluate long term investments and vice versa. This practice can be particularly critical in the context of pensions or health insurance, where long-term claims are expected and should be hedged. Long term investment are also effected by uncertainties of economic nature, as the uncertainty on interest rates. To capture both these situations in the same risk measurement we propose to work with fully dynamic risk cash non-additive risk measures. For this we propose the extension of shortfall risk measures or what we call the qh-entropic risk measures, within the BSDE framework

• Nunno, Giulia Di (2025). Memory and roughness in SVV models: Characteristics, pricing, hedging. doi: https:/www.londonmathfinance.org.uk/2025-seminars. Fulltekst i vitenarkiv Vis sammendrag

  SVV models, or Sandwiched Volterra Volatility models, are a class of dynamics able to capture both the long memory and the rough aspects of volatility as well as complying with several typical stylised features of volatilities. At the same time, they are treatable enough to allow for a rigorous and explicit analysis of pricing and hedging. We present this class of models with their features and potentials. We discuss pricing and hedging of financial derivatives in this framework. Specifically, we focus on the quadratic hedging approach, providing insight on the computational challenges related to the optimal strategy. The explicit optimal hedging solution is identified by means of the non-anticipating derivative, while its numerical counterpart is articulated by means of Markovian approximations.

• Lozano, Pere Diaz & Nunno, Giulia Di (2025). Deep Operator BSDEs for solution operators. doi: https:/sites.google.com/view/sure-ai-lsc2025/home?authuser=1. Fulltekst i vitenarkiv Vis sammendrag

  Motivated by dynamic risk measures and conditional g-expectations, in this presentation we present the ideas behind a method to approximate the solution operators. The main ingredients for this are the Wiener chaos decomposition coupled with the classical Euler scheme for BSDEs. Numerical examples implemented with neural networks are presented.

• Nunno, Giulia Di (2025). Challenges in risk measurement: horizon risk and intererst rates uncertainty. Fulltekst i vitenarkiv Vis sammendrag

  Different types of risks require different forms of evaluation. The properties of risk measures have to be carefully selected when taking environmental and financial risks into account. We review some of these properties commenting on their role in the analysis. Furthermore, we identify some underlying challenges, particularly evident when different time horizons have to be considered and when investments are scheduled in long terms. This leads to the indentification of horizon risk which has to be coupled with interest rates uncertainty.

• Nunno, Giulia Di (2025). Utility maximisation and change of variable formulas for time-changed dynamics. Fulltekst i vitenarkiv Vis sammendrag

  We derive novel change of variable formulas for stochastic integrals w.r.t. a time-changed Brownian motion where we assume that the time-change is a general increasing stochastic process with finitely many jumps in a bounded set of the positive half-line and is independent of the Brownian motion. As an application we consider the problem of maximising the expected utility of the terminal wealth in a semimartingale setting, where the semimartingale is written in terms of a time-changed Brownian motion and a finite variation process. To solve this problem, we use an initial enlargement of filtration and our change of variable formulas to shift the problem to a maximisation problem under the enlarged filtration for models driven by a Brownian motion and a finite variation process. The latter problem can be solved by using martingale properties. Then applying again a change of variable formula, we derive the optimal strategy for the original problem for a power utility and for a logarithmic utility.

• Nunno, Giulia Di (2025). Horizon risk in risk measuring. doi: https:/www.unibocconi.it/en/news/bocconi-hosts-8th-symposium-games-and-decisions-reliability-and-risk. Fulltekst i vitenarkiv Vis sammendrag

  Horizon risk is introduced as the assessing the exposure by a risk measure that is not adequate to the actual time horizon of the position. We consider the particularly the contest of investments in economics and finance. We clarify that also dynamic risk measures are subject to horizon risk, so we propose to work with fully-dynamic risk measures. We analyse and construct dynamic risk measures that appropriately take horizon risk into account. We also combine horizon risk with other uncertainties of future market scenarios, such as interest rates uncertainty, thus we extend the arguments and suggest the use of a cash non-additive version of fully- dynamic risk measures. We construct such risk measures via backward stochastic differential equations or via shortfall-type representation. As illustration, we introduce the class of hq-entropic risk measures.

• Nunno, Giulia Di (2025). Horizon risk and interest rates uncertainty in dynamic risk measuring. doi: https:/rt-matrisk2025.sciencesconf.org. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2025). Lifting of Volterra SDEs and optimal control. doi: https:/www.unipr.it/node/109533. Fulltekst i vitenarkiv Vis sammendrag

  We consider a stochastic optimal control problem for Volterra type forward dynamics, which are typically non-Markovian in nature. A direct application of dynamic pro- gramming is not feasible. So we propose to lift the problem to infinite dimensions, where we can recover Markovianity. For this we work with Banach-valued processes. In this framework we tackle the optimal control problem by studying the corresponding Hamilton-Jacobi-Bellman equation, which can be formulated thanks to the appropriate lift. Our techniques make use of Malliavin calculus in UMD Banach spaces. Finally, we reconnect the solution of the infinite dimensional problem to the original finite dimen- sional one.

• Nunno, Giulia Di (2025). Horizon risk and interest rates uncertainty in dynamic risk measuring. doi: https:/qmf2025.exordo.com. Fulltekst i vitenarkiv Vis sammendrag

  After last year's successful restart of this conference series, Quantitative Methods in Finance (QMF) 2025 will be held at the University of Technology Sydney (UTS) from 16 to 19 December 2025, bringing together leading experts in the Quantitative Finance Industry and Academia. In addition, the conference proper will be preceded by a pre-conference workshop on 15 December, Frontiers of Computational Finance, focusing on advances in applications of quantum computing and machine learning to finance. Furthermore, this year we have coordinated our conference scheduling with the workshop Climate Finance & Risk 2025, hosted separately to QMF by the Macquarie University Business School from 10 to 12 December. Focus This year the focus will be on Financing the Energy Transition, Pensions, Insurance, Machine Learning, Interest Rate Term Structure, Risk Measurement, Benchmark Approach, and other areas of Quantitative Finance. Plenary speakers include: Yacine Ait-Sahalia, Mesias Alfeus, Alexandre Antonov, Nan Chen, Christa Cuchiero, Giulia Di Nunno, Claudio Fontana, Masaaki Fukasawa, Martino Grasselli, Paolo Guasoni, Sebastian Jaimungal, Constantinos Kardaras, Martin Larsson, Sébastien Lleo, Andrea Macrina, Alexander Novikov, Jan Obloj, Marek Rutkowski, Thorsten Schmidt, Michael Schmutz, Pavel Shevchenko, and David Skovmand.

• Nunno, Giulia Di (2025). Horizon risk: what it is and how to tackle it. doi: https:/landing.cafoscarichallengeschool.it/docra/#aims. Fulltekst i vitenarkiv
• Lozano, Pere Diaz (2025). Deep operator BSDE: A numerical scheme to approximate solution operators. Fulltekst i vitenarkiv Vis sammendrag

  Motivated by dynamic risk measures and conditional g-expectations, in this work we propose a numerical method to approximate the solution operator given by a backward stochastic differential equation (BSDE). The main ingredients for this are the Wiener chaos decomposition and the classical Euler scheme for BSDEs. We show convergence of this scheme under very mild assumptions, and provide a rate of convergence in more restrictive cases. We then implement it using neural networks, and we present several numerical examples where we can check the accuracy of the method.

• Baños, David & Bernabe, Rosemarie de La Cruz (2025). En akademisk og medisinsk skandale. Forskersonen.no. Fulltekst i vitenarkiv Vis sammendrag

  DEBATT: At seniorforskere med betydelig makt og innflytelse ikke ivaretar pasienters og forskningsdeltakeres rettigheter, og ikke anerkjenner sakens alvor, er dypt urovekkende.

• Ortiz-Latorre, Salvador (2025). Robustness of quasi-linear SPDEs with respect to the driving noise. Fulltekst i vitenarkiv
• Baños, David; Marstein, Alma; Patrascu, Monica; Hartmann, Anine; Bromseth, Janne C. Haugnes & Korbmacher, Max (2025). Response to authors’ reply: persistent concerns regarding Nyquist et al. from 15 July 2024. European Child and Adolescent Psychiatry. ISSN 1018-8827. doi: 10.1007/s00787-025-02883-7. Fulltekst i vitenarkiv
• Baños, David; Marstein, Alma; Patrascu, Monica; Hartmann, Anine; Bromseth, Janne & Korbmacher, Max (2025). Concerns About Nyquist, Torgersen, David, Diseth, Gulbrandsen and Wæhre's “Treatment Trajectories Among Children and Adolescents”. Acta Paediatrica. ISSN 0803-5253. doi: 10.1111/apa.70375. Fulltekst i vitenarkiv
• Benth, Fred Espen (2025). Structure-informed learning and interpretability -- the case of energy markets. Fulltekst i vitenarkiv
• Benth, Fred Espen (2025). Hva kan en matematiker bidra med i risikohåndtering? Fulltekst i vitenarkiv
• Benth, Fred Espen (2025). Weather derivatives. Fulltekst i vitenarkiv
• Benth, Fred Espen (2025). Hedging weather risk in energy markets. Fulltekst i vitenarkiv
• Benth, Fred Espen (2025). Using informed neural networks to price power options. Fulltekst i vitenarkiv
• Benth, Fred Espen (2025). Using informed neural networks to price options on power forward -- and some more on forwards. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2025). Utility maximisation and change of variable formulas for time-changed dynamics. Fulltekst i vitenarkiv
• Baños, David; Marstein, Alma; Patrascu, Monica; Hartmann, Anine; Bromseth, Janne C. Haugnes & Korbmacher, Max (2025). Correction to: Concerns about Nyquist et al., published 15 July 2024 (European Child & Adolescent Psychiatry, (2025), 10.1007/s00787-025-02649-1). European Child and Adolescent Psychiatry. ISSN 1018-8827. doi: 10.1007/s00787-025-02668-y. Fulltekst i vitenarkiv
• Skogvoll, Vidar; Lindstrøm, Tom Louis; Benth, Fred Espen & Haraldsrud, Andreas (2024). Real læring - Realistfabrikken: Hvordan lage matematiker. [Radio]. Podcast. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2024). Fully-dynamic risk measures via (Volterra) BSDEs: horizon risk and interest rate uncertainty. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2024). Malliavin calculus in SVV models: Pricing and Power law. Fulltekst i vitenarkiv
• Lindstrøm, Tom Louis (2024). Historien om å løse ligninger. Fulltekst i vitenarkiv
• Viole, Isabelle; Greevenbroek, Koen van & Cheng, Claudia Siew Wan (2024). Can Norway save the European Union's hydrogen ambition for 2030? Fulltekst i vitenarkiv
• Nunno, Giulia Di (2024). The complexity of risk measurement. Fulltekst i vitenarkiv
• Melteig, Elina & Nunno, Giulia Di (2024). Hva skjer hvis alle sparer samtidig? [Internet]. forskning.no. Fulltekst i vitenarkiv
• Melteig, Elina & Nunno, Giulia Di (2024). Hva skjer hvis alle sparer samtidig? [Internet]. Titan.uio.no. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2024). Lifting of Volterra processes: Optimal control in UMD Banach spaces. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2024). A tour in Sandwiched Volterra Volatility models. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2024). Fully-dynamic risk measures: horizon risk and interest rate uncertainty. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2024). Fully-dynamic risk measures: horizon risk and interest rate uncertainty. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2024). Sandwiched SDEs and volatility modelling. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2024). Sandwiched Volterra volatility models, power law and option pricing. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2024). Deep Operator BSDE. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2024). Deep Operator BSDE. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2024). On Sandwich Volterra Volatility models. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2024). On the power law in a class of sandwiched Volterra volatility models. Fulltekst i vitenarkiv
• Font, Oriol Zamora (2024). Pricing VIX options under the Heston-Hawkes stochastic volatility model. Fulltekst i vitenarkiv
• Yurchenko-Tytarenko, Anton (2024). Quadratic hedging in SVV-model. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2024). Deep Operator BSDE. Fulltekst i vitenarkiv
• Ortiz-Latorre, Salvador (2024). Deep learning methods for the stochastic filtering problem. Fulltekst i vitenarkiv
• Savku, Emel (2023). Stochastic Maximum Principle For A Constraint Nonzero-Sum Game Application:Bancassurance. Fulltekst i vitenarkiv
• Melteig, Elina & Grochowicz, Aleksander (2023). Strømmarkedet: Hva skjer hvis vi kobler Nord- og Sør-Norge? [Internet]. forskning.no, tu.no, titan.uio.no. Fulltekst i vitenarkiv
• Huseby, Arne Bang (2023). Domination and multistate systems. Fulltekst i vitenarkiv Vis sammendrag

  Domination theory has been studied extensively in the context of binary monotone systems, where the structure function is a sum of products of the component states, and with coefficients given by the signed domination function. Using e.g., matroid theory, many useful properties of the signed domination function can be derived. In this paper we show how some of these results can be extended to multistate systems. The signed domination function is an invariant determined by the poset generated by the minimal path vectors. Two systems with isomorphic posets also have the same signed domination function. We derive an explicit formula for the domination using Möbius functions. Furthermore, we obtain a recursive formula for the signed domination function, referred to as the signed domination theorem. Both these results are generalizations of the corresponding results for binary monotone systems.

• Grochowicz, Aleksander & Zeyringer, Marianne (2023). Managing the variability in time and geography of renewable power generation. Fulltekst i vitenarkiv
• Grochowicz, Aleksander (2023). The role of historical weather data and how that can be used for a more resilient energy transition. Fulltekst i vitenarkiv
• Grochowicz, Aleksander & Zeyringer, Marianne (2023). Managing the variability in time and space of renewable power generation. Fulltekst i vitenarkiv
• Font, Oriol Zamora; Baños, David & Ortiz-Latorre, Salvador (2023). Heston-Hawkes Stochastic Volatility Model: Change of Measure and Forward Variance. Fulltekst i vitenarkiv
• Grochowicz, Aleksander; Greevenbroek, Koen van; Benth, Fred Espen & Zeyringer, Marianne (2023). Intersecting Near-Optimal Spaces for Policy Information. Fulltekst i vitenarkiv
• Font, Oriol Zamora; Baños, David & Ortiz-Latorre, Salvador (2023). Heston-Hawkes stochastic volatility model: change of measure and forward variance. Fulltekst i vitenarkiv
• Grochowicz, Aleksander (2023). Spatio-temporal smoothing and dynamics of different electricity flexibility options. Fulltekst i vitenarkiv
• Greevenbroek, Koen van; Zeyringer, Marianne; Noll, Josef; Grochowicz, Aleksander & Vågerö, Oskar (2023). Kjernekraft er ikkje naudsynt for Noreg. Det vil heller ikkje lønna seg. Aftenposten (morgenutg. : trykt utg.). ISSN 0804-3116. Fulltekst i vitenarkiv
• Grochowicz, Aleksander; Greevenbroek, Koen van & Bloomfield, Hannah C. (2023). Identifying weather stress events from power system optimisation outputs. Fulltekst i vitenarkiv
• Grochowicz, Aleksander (2023). Identifying weather stress events from power system optimisation outputs. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2023). Lifting of Volterra processes: Optimal control in UMD Banach spaces. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2023). Sandwiched Volterra volatility models and option pricing. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2023). Optimal control for Volterra type dynamics. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2023). Sandwiched Volterra volatility models and hedging. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2023). Horizon risk and fully dynamic risk measures. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2023). Research on Risk Measures. Fulltekst i vitenarkiv
• Ortiz-Latorre, Salvador (2023). Strong solutions for singular SDEs driven by fractional Brownian motion. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2023). Sandwiched Volterra volatility models and hedging. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2023). Dynamic risk assessment, horizon risk, and interest rate uncertainty. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2023). Pricing in sandwiched Volterra volatility models. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2023). Horizon risk and fully dynamic risk measures. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2023). Sandwiched Volterra volatility models and hedging. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2023). Horizon risk and dynamic risk assessment. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2023). On stochastic control for time changed Lévy dynamics. Fulltekst i vitenarkiv
• Ortiz-Latorre, Salvador (2023). Pathwise approximations for the solution of the non-linear filtering problem. Fulltekst i vitenarkiv
• Greevenbroek, Koen van; Grochowicz, Aleksander & Bloomfield, Hannah C. (2023). Identifying Weather Stress Events from Power System Optimisation Outputs. Fulltekst i vitenarkiv
• Eggen, Mari Dahl; Midtfjord, Alise Danielle; Vorobeva, Ekaterina; Benth, Fred Espen; Hupe, Patrick & Brissaud, Quentin [Vis alle 10 forfattere av denne artikkelen] (2023). Using a machine learning and stochastics-founded model to provide near real-time stratospheric polar vortex diagnostics based on high-latitude infrasound data. Fulltekst i vitenarkiv Vis sammendrag

  Acoustic waves below the frequency limit of human hearing - infrasound - can travel for thousands of kilometres in the atmosphere. The global propagation signature of infrasound is highly sensitive to the wind structure of the stratosphere. This work exploits processed continuous data from three high-latitude infrasound stations to characterize an aspect of the stratospheric polar vortex. Concretely, a mapping is developed which takes the infrasound data from these three stations as input and outputs an estimate of the polar cap zonal mean wind averaged over 60-90 degrees in latitude at the 1 hPa pressure level. This stratospheric diagnostic information is relevant to, for example, sudden stratospheric warming assessment and sub-seasonal prediction. The considered acoustic data is within a low-frequency regime globally dominated by so-called microbarom infrasound, which is continuously radiated into the atmosphere due to nonlinear interaction between counter-propagating ocean surface waves. We trained a stochastics-based machine learning model (delay-SDE-net) to map between a time series of five years (2014-2018) of processed infrasound data and the ERA5 (reanalysis-based) daily average polar cap wind at 1 hPa for the same period. The ERA5 data was hence treated as ground-truth. In the prediction, the delay-SDE-net utilizes time-lagged inputs and their dependencies, as well as the day of the year to account for seasonal differences. In the validation phase, the input was the 2019 and 2020 infrasound time series, and the model inference results in an estimate of the daily average polar cap wind time-series. This result was then compared to the ERA5 representation of the stratospheric diagnostic time-series for the same period. The applied machine learning model is based on stochastics and allows for an interpretable approach to estimate the aleatoric and epistemic prediction uncertainties. It is found that the mapping, which is only informed of the trained model, the day of year, and the infrasound data from three stations, generates a 1 hPa polar cap average wind estimate with a prediction error standard deviation of around 10 m/s compared to ERA5. Focus should be put on the winter months because this is when the coupling between the stratosphere and the troposphere can mostly influence the surface conditions and provide additional prediction skill, in particular during strong and weak stratospheric polar vortex regimes. The infrasound data is available in real-time, and we discuss how the developed approach can be extended to provide near real-time stratospheric polar vortex diagnostics.

• Greevenbroek, Koen van; Grochowicz, Aleksander; Zeyringer, Marianne; Noll, Josef & Vågerö, Oskar (2023). Vi treng uavhengig forsking, ikkje drøymetenking om kjernekraft. Aftenposten (morgenutg. : trykt utg.). ISSN 0804-3116. Fulltekst i vitenarkiv
• Yurchenko-Tytarenko, Anton (2023). Volterra sandwiched volatility model: Markovian approximation and hedging. Fulltekst i vitenarkiv
• Yurchenko-Tytarenko, Anton (2023). Volterra sandwiched volatility model: Markovian approximation and hedging. Fulltekst i vitenarkiv
• Yurchenko-Tytarenko, Anton (2023). CIR process and Skorokhod problems]{Cox-Ingersoll-Ross process and Skorokhod problems. Fulltekst i vitenarkiv
• Yurchenko-Tytarenko, Anton (2023). Hedging in Volterra volatility models. Fulltekst i vitenarkiv
• Yurchenko-Tytarenko, Anton (2023). Volterra sandwiched volatility model: Markovian approximation and hedging. Fulltekst i vitenarkiv
• Font, Oriol Zamora; Baños, David & Ortiz-Latorre, Salvador (2023). Change of measure in a Heston-Hawkes stochastic volatility model. Fulltekst i vitenarkiv
• Yurchenko-Tytarenko, Anton (2023). Parameter estimation in rough Bessel model. Fulltekst i vitenarkiv
• Yurchenko-Tytarenko, Anton (2023). Power law in SVV model. Fulltekst i vitenarkiv
• Lozano, Pere Diaz (2023). Neural SDEs for Conditional Time Series Generation using the Signature Wasserstein-1 metric. Fulltekst i vitenarkiv
• Ortiz-Latorre, Salvador (2023). SPDE bridges with observation noise and their spatial approximation. Fulltekst i vitenarkiv
• Ortiz-Latorre, Salvador (2023). Strong solutions for singular SDEs driven by fractional Brownian motion. Fulltekst i vitenarkiv
• Ortiz-Latorre, Salvador (2023). Time discretizations for the nonlinear filtering problem. Fulltekst i vitenarkiv
• Ortiz-Latorre, Salvador (2023). Robust approximations for the solution of the non-linear filtering problem. Fulltekst i vitenarkiv
• Ortiz-Latorre, Salvador (2023). Robust discretizations for the solution of the non-linear filtering problem. Fulltekst i vitenarkiv
• Font, Oriol Zamora; Baños, David & Ortiz-Latorre, Salvador (2023). Heston-Hawkes stochastic volatility model: Change of measure and Thiele's PIDE. Fulltekst i vitenarkiv
• Petersson, Andreas Erik (2023). The HEat modulated Infinite DImensional Heston (HEIDIH) model and its numerical approximation. Fulltekst i vitenarkiv
• Petersson, Andreas Erik (2023). Noise Simulation in Finite Element Methods for SPDE Approximation. Fulltekst i vitenarkiv
• Petersson, Andreas Erik (2023). SPDE bridges with observation noise and their spatial approximation. Fulltekst i vitenarkiv
• Petersson, Andreas Erik (2023). Piecewise linear interpolation of noise in finite element approximations of parabolic SPDEs. Fulltekst i vitenarkiv
• Savku, Emel (2023). A Nonzero-Sum Regime-Switching Stochastic Differential Game Application with Constraints. Fulltekst i vitenarkiv
• Savku, Emel (2023). A Stochastic Maximum Principle Approach for a Nash Equilibrium of a Nonzero-Sum Game. Fulltekst i vitenarkiv
• Benth, Fred Espen (2022). Pricing options on flow forwards by neural networks in Hilbert space. Fulltekst i vitenarkiv
• Savku, Emel (2022). An Application of Nonzero-Sum Stochastic Differential Games in Finance. Fulltekst i vitenarkiv
• Savku, Emel (2022). An Application of Stochastic Differential Games with Lagrange Multipliers: Bancassurance. Fulltekst i vitenarkiv
• Dordevic, Jasmina (2022). "On reflected backward stochastic differential equations driven by time-changed Lévy noise”. Fulltekst i vitenarkiv
• Dordevic, Jasmina (2022). “Time-changed Lévy process and application of stopping problems to RBSDEs“. Fulltekst i vitenarkiv
• Dordevic, Jasmina (2022). “Modelling of epidemics with time-changed Lévy process”. Fulltekst i vitenarkiv
• Dordevic, Jasmina (2022). “Backward stochastic differential equations”. Fulltekst i vitenarkiv
• Dordevic, Jasmina (2022). “Interference of time-change Lévy noises on characterisation of reflected backward stochastic differential equations“. Fulltekst i vitenarkiv
• Dordevic, Jasmina (2022). “Reflected Backward Stochastic Differential Equations with Time-change Lévy noises.”. Fulltekst i vitenarkiv
• Petersson, Andreas Erik (2022). Interpolation of noise in the finite element approximation of the HEIDIH model. Fulltekst i vitenarkiv
• Petersson, Andreas Erik (2022). SPDE bridges with observation noise and their spatial approximation. Fulltekst i vitenarkiv
• Petersson, Andreas Erik (2022). Interpolation of noise in finite element approximation of stochastic PDEs. Fulltekst i vitenarkiv
• Petersson, Andreas Erik (2022). Numerical approximation of the heat modulated infinite dimensional Heston model. Fulltekst i vitenarkiv
• Petersson, Andreas Erik (2022). Numerical approximation of the heat modulated infinite dimensional Heston model. Fulltekst i vitenarkiv
• Petersson, Andreas Erik (2022). ​Interpolation of noise in finite element approximation of stochastic PDEs. Fulltekst i vitenarkiv
• Ortiz-Latorre, Salvador (2022). Deep learning methods for the stochastic filtering problem. Fulltekst i vitenarkiv
• Eggen, Mari Dahl; Dahl, Kristina Rognlien; Näsholm, Sven Peter & Mæland, Steffen (2022). Stochastic Modeling of Stratospheric Temperature. Fulltekst i vitenarkiv Vis sammendrag

  This study suggests a stochastic model for time series of daily zonal (circumpolar) mean stratospheric temperature at a given pressure level. It can be seen as an extension of previous studies which have developed stochastic models for surface temperatures. The proposed model is a combination of a deterministic seasonality function and a Lévy-driven multidimensional Ornstein–Uhlenbeck process, which is a mean-reverting stochastic process. More specifically, the deseasonalized temperature model is an order 4 continuous-time autoregressive model, meaning that the stratospheric temperature is modeled to be directly dependent on the temperature over four preceding days, while the model’s longer-range memory stems from its recursive nature. This study is based on temperature data from the European Centre for Medium-Range Weather Forecasts ERA-Interim reanalysis model product. The residuals of the autoregressive model are well represented by normal inverse Gaussian-distributed random variables scaled with a time-dependent volatility function. A monthly variability in speed of mean reversion of stratospheric temperature is found, hence suggesting a generalization of the fourth-order continuous-time autoregressive model. A stochastic stratospheric temperature model, as proposed in this paper, can be used in geophysical analyses to improve the understanding of stratospheric dynamics. In particular, such characterizations of stratospheric temperature may be a step towards greater insight in modeling and prediction of large-scale middle atmospheric events, such as sudden stratospheric warming. Through stratosphere–troposphere coupling, the stratosphere is hence a source of extended tropospheric predictability at weekly to monthly timescales, which is of great importance in several societal and industry sectors.

• Amine, Oussama; Mansouri, Abdol-Reza & Proske, Frank Norbert (2022). Well-posedness of the Deterministic Transport Equation with Singular Velocity Field Perturbed along Fractional Brownian Paths. Fulltekst i vitenarkiv
• Amine, Oussama; Mansouri, Abdol-Reza & Proske, Frank Norbert (2022). Well-posedness of the Deterministic Transport Equation with Singular Velocity Field Perturbed along Fractional Brownian Paths. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2022). Sandwiched SDEs with unbounded drift, Hölder noises and stochastic volatility modelling. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2022). Horizon risk and fully-dynamic risk measures. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2022). Fully-dynamic risk measures: time-consistency, horizon risk, and relations with BSDEs and BSVIEs. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2022). Optimal control of Volterra type equations: from finite to infinite dimensions and return. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2022). On time changed Lévy noises in modelling, dynamics and control. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2022). Stochastic games for Volterra time-changed Lévy dynamics. Fulltekst i vitenarkiv
• Nunno, Giulia Di; Natalini, Roberto & Luca, Perri (2022). La matematica unisce. [Internet]. https://www.youtube.com/watch?v=-Sj-3w0GXvw. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2022). Fully-dynamic risk measures and horizon risk. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2022). Stochastic games for Volterra time-changed Lévy dynamics. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2022). Sandwiched SDEs with unbounded drift driven by Hölder noises. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2022). Stochastic games for Volterra time-changed Lévy dynamics. Fulltekst i vitenarkiv
• Font, Oriol Zamora; Baños, David & Ortiz-Latorre, Salvador (2022). Change of measure in a Heston-Hawkes stochastic volatility model. Fulltekst i vitenarkiv
• Ortiz-Latorre, Salvador (2022). Machine learning applied to the stochastic filtering problem. Fulltekst i vitenarkiv
• Ortiz-Latorre, Salvador (2022). High order discretizations to the nonlinear filtering problem. Fulltekst i vitenarkiv
• Savku, Emel (2022). A Constrained Nonzero-Sum Stochastic Differential Game Application. Fulltekst i vitenarkiv
• Savku, Emel (2022). An Application of Stochastic Differential Games with Lagrange Multipliers:Bancassurance. Fulltekst i vitenarkiv
• Grochowicz, Aleksander (2022). Intersecting near-optimal spaces for more resilience against weather variability. Fulltekst i vitenarkiv
• Yurchenko-Tytarenko, Anton (2022). Volterra Sandwiched Volatility models. Fulltekst i vitenarkiv
• Benth, Fred Espen (2022). Modellering av risiko i energisystemer. Fulltekst i vitenarkiv
• Yurchenko-Tytarenko, Anton (2022). Volterra sandwiched volatility model: Markovian approximation and hedging. Fulltekst i vitenarkiv
• Yurchenko-Tytarenko, Anton (2022). Volterra Sandwiched Volatility models. Fulltekst i vitenarkiv
• Savku, Emel (2022). A constrained stochastic differential game application: Bancassurance. Fulltekst i vitenarkiv
• Savku, Emel (2022). A Constrained Nonzero-Sum Game Application: Bancassurance. Fulltekst i vitenarkiv
• Greevenbroek, Koen van & Grochowicz, Aleksander (2022). Effect of different weather years on policymakers’ decision space. Fulltekst i vitenarkiv
• Grochowicz, Aleksander; Greevenbroek, Koen van; Benth, Fred Espen & Zeyringer, Marianne (2022). Intersecting near-optimal spaces for robust energy systems. Fulltekst i vitenarkiv
• Harang, Fabian Andsem (2022). Pathwise regularization by noise for SDEs and SPDEs with multiplicative noise. Fulltekst i vitenarkiv
• Savku, Emel (2022). An Application of Markov Regime-Switching Models: Bancassurance. Fulltekst i vitenarkiv
• Zeyringer, Marianne; Benth, Fred Espen; Roithner, Maximilian; Grochowicz, Aleksander & Sirotko-Sibirskaya, Natalia (2022). Climate-resilient net-zero energy system design. Fulltekst i vitenarkiv
• Benth, Fred Espen (2022). Klima og vær - data og risiko. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2021). Maximum principles for stochastic time-changed Volterra games. Fulltekst i vitenarkiv
• Nunno, Giulia Di & Yurchenko-Tytarenko, Anton (2021). Rough volatility: SDE with unbounded drift driven by Hölder continuous noise. Fulltekst i vitenarkiv
• Nunno, Giulia Di; Benth, Fred Espen & Simonsen, Iben Cathrine (2021). Infinite dimensional Heston model and sensitivity analysis. Fulltekst i vitenarkiv
• Dahl, Kristina Rognlien & Eyolfsson, Heidar (2021). Self-exciting jump processes as deterioration models. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2021). Sensitivity analysis in the infinite dimensional Heston model. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2021). Sensitivity analysis in the infinite dimensional Heston model. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2021). Optimal portfolios in markets with memory. Fulltekst i vitenarkiv
• Giordano, Michele & Yurchenko-Tytarenko, Anton (2021). Optimal control in linear stochastic advertising models with memory. Fulltekst i vitenarkiv
• Dordevic, Jasmina (2021). “Backward stochastic differential equations with interaction”. Fulltekst i vitenarkiv
• Dordevic, Jasmina (2021). “Stochastic analysis of spread & preventions in case of SARS-CoV-2 virus”. Fulltekst i vitenarkiv
• Dordevic, Jasmina (2021). “Clark representation for random measures”. Fulltekst i vitenarkiv
• Benth, Fred Espen & Zeyringer, Marianne (2021). Uncertain energy systems. Fulltekst i vitenarkiv
• Eggen, Mari Dahl (2021). Time Series and CNNs - Three articles on the topic. Fulltekst i vitenarkiv
• Benth, Fred Espen (2021). Pathwise Gaussian Volterra processes in Hilbert space. Fulltekst i vitenarkiv
• Benth, Fred Espen & Zeyringer, Marianne (2021). Uncertain Energy Systems. Fulltekst i vitenarkiv
• Savku, Emel (2021). An Application of Stochastic Maximum Principle with Regimes and Memory. Fulltekst i vitenarkiv
• Petersson, Andreas Erik (2021). Approximation of SPDE covariance operators by finite elements: A semigroup approach. Fulltekst i vitenarkiv
• Petersson, Andreas Erik (2021). SPDE bridges with observation noise and their spatial approximation. Fulltekst i vitenarkiv
• Petersson, Andreas Erik (2021). Approximation of SPDE covariance operators by finite elements: A semigroup approach. Fulltekst i vitenarkiv
• Midtfjord, Alise Danielle (2021). The Fundamentals of AI. Fulltekst i vitenarkiv Vis sammendrag

  What's Artificial Intelligence (AI)? What's the difference between AI and machine learning (ML), as they're sometimes used interchangeably? What's the current state of AI? What is AI capable of these days and what not? What are the societal impact and implications of AI? These are questions we will discuss in the upcoming event co-organized by Pint of Science Norway (http://pintofscience.no/) and Oslo AI (http://oslo.city.ai/). We've gathered a panel of experts coming from academia and industry. They will demystify AI for you and answers all your questions.

• Nunno, Giulia Di; Mishura, Yuliya & Yurchenko-Tytarenko, Anton (2021). Volterra Sandwiched Volatility Model. Fulltekst i vitenarkiv
• Nunno, Giulia Di; Mishura, Yuliya & Yurchenko-Tytarenko, Anton (2021). Approximating expected value of an option with non-Lipschitz payoff in fractional Heston-type model. Fulltekst i vitenarkiv
• Nunno, Giulia Di; Mishura, Yuliya & Yurchenko-Tytarenko, Anton (2021). Volterra Sandwiched Volatility Model. Fulltekst i vitenarkiv
• Nunno, Giulia Di; Mishura, Yuliya & Yurchenko-Tytarenko, Anton (2021). Stochastic volatility modelling via sandwiched processes with Volterra noise. Fulltekst i vitenarkiv
• Nunno, Giulia Di; Mishura, Yuliya & Yurchenko-Tytarenko, Anton (2021). Volterra Sandwiched Volatility Model. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2021). Sensitivity analysis in the infinite dimensional Heston model. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2021). Infinite dimensional Heston model and sensitivity analysis. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2021). Optimal portfolios in markets with memory. Fulltekst i vitenarkiv
• Dordevic, Jasmina (2021). "Clark representation for random measures". Fulltekst i vitenarkiv
• Nunno, Giulia Di (2021). Sensitivity analysis in the infinite dimensional Heston model. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2021). Rough volatility: SDE with unbounded drift driven by Hölder continuous noise. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2021). Maximum principles for stochastic time-changed Volterra games. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2021). On time changed Lévy noises in modelling, dynamics and control. Fulltekst i vitenarkiv
• Benth, Fred Espen (2021). Hedging volumetric risk in renewable energy markets. Fulltekst i vitenarkiv
• Benth, Fred Espen & Schrader, Simon Elias (2021). Send krafta til Tyskland. Klassekampen. ISSN 0805-3839. Fulltekst i vitenarkiv
• Savku, Emel (2021). A Nonzero-sum Game Formulation for a Markov Regime-Switching Portfolio Strategy. Fulltekst i vitenarkiv
• Lindstrøm, Tom Louis (2021). Matematikk og virkelighet: Om å beregne det som ikke kan måles. Fulltekst i vitenarkiv
• Midtfjord, Alise Danielle (2021). A decision support system for safer airplane landings: predicting runway conditions using XGBoost and explainable AI. Fulltekst i vitenarkiv
• Eggen, Mari Dahl; Dahl, Kristina Rognlien; Näsholm, Sven Peter & Mæland, Steffen (2021). Stochastic modelling of stratospheric temperature. Fulltekst i vitenarkiv
• Benth, Fred Espen & Zeyringer, Marianne (2021). Er de rekordhøye strømprisene verdt det? Fulltekst i vitenarkiv
• Lindstrøm, Tom Louis (2021). Utforsking i matematikk: Trampoline eller hengemyr? Fulltekst i vitenarkiv
• Lindstrøm, Tom Louis (2021). Programmering i matematikkfaget. Fulltekst i vitenarkiv
• Lindstrøm, Tom Louis (2021). Implementation of Computational Thinking at the University of Oslo. Fulltekst i vitenarkiv
• Midtfjord, Alise Danielle (2021). A Machine Learning Approach to Safer Airplane Landings. Fulltekst i vitenarkiv
• Midtfjord, Alise Danielle (2021). A Machine Learning Approach to Assess Runway Conditions Using Weather Data. Fulltekst i vitenarkiv
• Savku, Emel (2021). Stochastic Optimal Control Techniques for a Regime-Switching Model with Applications in Finance. Fulltekst i vitenarkiv
• Yurchenko-Tytarenko, Anton (2021). Standard and fractional ROU processes as the limits of square roots of CIR processes. Fulltekst i vitenarkiv
• Dordevic, Jasmina (2021). Effect of Additive Perturbations on the Solution of Reflected Backward Stochastic Differential Equations. I Carpentieri, Bruno (Red.), Recent Developments in the Solution of Nonlinear Differential Equations. IntechOpen. ISSN 9781839686573. doi: 10.5772/intechopen.95872. Fulltekst i vitenarkiv Vis sammendrag

  This chapter has as a topic large class of general, nonlinear reflected backward stochastic differential equations with a lower barrier, whose generator, final condition as well as barrier process arbitrarily depend on a small parameter. The solutions of these equations which are obtained by additive perturbations, named the perturbed equations, are compared in the L p -sense, p ∈ ] 1 , 2 [ , with the solutions of the appropriate equations of the equal type, independent of a small parameter and named the unperturbed equations. Conditions under which the solution of the unperturbed equation is L p -stable are given. It is shown that for an arbitrary a > 0 there exists t a ≤ T , such that the L p -difference between the solutions of both the perturbed and unperturbed equations is less than a for every t ∈ t a T .

• Savku, Emel (2021). Stochastic Maximum Principle with Regimes and Memory. Fulltekst i vitenarkiv
• Savku, Emel (2021). Portfolio Strategies via Stochastic Differential Games with Regimes. Fulltekst i vitenarkiv
• Harang, Fabian Andsem (2021). Pathwise regularization by noise for SDEs and SPDEs with multiplicative noise. Fulltekst i vitenarkiv
• Harang, Fabian Andsem (2021). Log-modulated fractional Brownian motion and super rough volatility. Fulltekst i vitenarkiv
• Harang, Fabian Andsem (2021). Introduction to Rough Paths Theory. Fulltekst i vitenarkiv
• Savku, Emel (2021). Stochastic Differential Games via Dynamic Programming Principle with Regimes. Fulltekst i vitenarkiv
• Savku, Emel (2021). Stochastic Differential Games within the framework of Regime-Switches. Fulltekst i vitenarkiv
• Benth, Fred Espen (2021). Pathwise Gaussian Volterra processes in Hilbert space. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2020). Infiite dimensional Heston model: pricing and sensitivity analysis. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2020). Stochastic control for Volterra equations driven by time-changed noises. Fulltekst i vitenarkiv
• Dahl, Kristina Rognlien (2020). The SCROLLER project A Stochastic ContROL approach to machine Learning with applications to Environmental Risk models. Fulltekst i vitenarkiv
• Nunno, Giulia Di (2020). Stochastic control for Volterra equations driven by time-changed noises. Fulltekst i vitenarkiv
• Giordano, Michele & Nunno, Giulia Di (2020). Lifting of Volterra processes: optimal control and HJB equations. Fulltekst i vitenarkiv
• Dahl, Kristina Rognlien (2020). FBSDE games with delay & noisy memory. Fulltekst i vitenarkiv
• Harang, Fabian Andsem (2020). Infinitely regularizing paths, and regularization by noise. Fulltekst i vitenarkiv
• Amine, Oussama; Mansouri, Abdol-Reza & Proske, Frank Norbert (2020). Well-posedness of the Deterministic Transport Equation with Singular Velocity Field Perturbed along Fractional Brownian Paths. Fulltekst i vitenarkiv
• Lozano, Pere Diaz; Nunno, Giulia Di & Ortiz-Latorre, Salvador (2026). Backward stochastic equations and martingale modelling. Universitetet i Oslo. Fulltekst i vitenarkiv Vis sammendrag

  This thesis is concerned with numerical methods and regularity properties for backward stochastic equations, together with martingale modelling in mathematical finance. The main tools are Wiener chaos expansions and Malliavin calculus. The first part focuses on backward stochastic differential equations (BSDEs). We develop numerical methods that do not require the terminal condition to be given as a function of a forward diffusion, as is often assumed in the literature. Instead, the terminal condition is represented through its Wiener chaos expansion. This leads to two related approaches: an operator-based Euler scheme that approximates the map sending terminal conditions to BSDE solutions, and a chaos-based Euler scheme in which conditional expectations and martingale integrands are computed through finite chaos projections. Convergence results are obtained for both methods. In the second part, we apply the same Wiener chaos point of view to financial modelling. Starting from the no-arbitrage requirement that discounted prices should be martingales under a risk-neutral measure, we construct an over-parameterized square-integrable martingale model by parameterizing its terminal value through a finite Wiener chaos expansion. The resulting model is arbitrage-free by construction, dense in the space of square-integrable martingales, and can be efficiently calibrated to implied volatility surfaces. The final part of the thesis studies backward stochastic Volterra integral equations (BSVIEs). We establish Hölder-type regularity estimates for the martingale integrand, both for the original BSVIE and for a system of approximating BSDEs. These estimates are then used to prove a convergence rate for an explicit Euler-type approximation of the BSVIE.

• Grochowicz, Aleksander (2024). Resilience and weather uncertainty in energy system models. Universitetet i Oslo. Fulltekst i vitenarkiv
• Grochowicz, Aleksander; Heineken, Daniel & Wennberg, Sondre (2021). The reliability of wind power in the Longyearbyen area. Universitetssenteret på Svalbard. Fulltekst i vitenarkiv Vis sammendrag

  This report in a white paper format was written as part of the course AGF-353/853 "Sustainable Arctic Energy Exploration and Development" at the University Centre in Svalbard (UNIS). --- Within this decade the current energy system in Longyearbyen has to be revolutionised. Right now, it relies entirely on fossil fuels, but in the future it is envisioned to be a front-runner of renewable systems in the Arctic. Due to the polar night and restricted renewable resources, the reliability of wind power is of crucial importance in this transition. Additionally, the permafrost and remote location demand high robustness and simple maintenance. In this report, we analyse on-shore wind potential in three different locations over the previous decade, including an evaluation of stability and extreme events in Svalbard's harsh climate. To ensure long-term success of this transition, we look at interannual, seasonal, and sub-monthly variations that should be included in suitability assessments. Based on our analysis, disregarding this could lead to installations that can jeopardise the stability of the system. With generated input data that we feed into an energy systems model, we find that if one considers interannual variability and a set of different locations, one can ensure a viable and reliable renewable energy system for Longyearbyen, with on-shore wind as a key component.

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