Frank Norbert Proske

• 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.

• 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.

• 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
• 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.

• 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).

• 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 .

• 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.

• 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.

• 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.

• 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
• 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.

• 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.

• 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.

• 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.

• 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
• 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
• 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
• 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.

• 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].

• 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
• 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
• 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.

• Agram, Nacira; Haadem, Sven; Øksendal, Bernt Karsten & Proske, Frank Norbert (2021). Optimal stopping, randomized stopping and singular control with general information flow. Theory of Probability and its Applications. ISSN 0040-585X. 66(4), s. 760–773. doi: 10.4213/tvp5514. Fulltekst i vitenarkiv
• Baños, David; Ortiz-Latorre, Salvador; Pilipenko, Andrey & Proske, Frank Norbert (2021). Strong Solutions of Stochastic Differential Equations with Generalized Drift and Multidimensional Fractional Brownian Initial Noise. Journal of theoretical probability. ISSN 0894-9840. doi: 10.1007/s10959-021-01084-7. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we prove the existence of strong solutions to an stochastic differential equation with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters H<12. Here, the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a “local time variational calculus” argument.

• Amine, Oussama; Baños, David & Proske, Frank Norbert (2020). Regularity properties of the stochastic flow of a skew fractional Brownian motion. Infinite Dimensional Analysis Quantum Probability and Related Topics. ISSN 0219-0257. 23(1). doi: 10.1142/S0219025720500058. Fulltekst i vitenarkiv Vis sammendrag

  In this paper we prove, for small Hurst parameters, the higher-order differentiability of a stochastic flow associated with a stochastic differential equation driven by an additive multi-dimensional fractional Brownian noise, where the bounded variation part is given by the local time of the unknown solution process. The proof of this result relies on Fourier analysis-based variational calculus techniques and on intrinsic properties of the fractional Brownian motion.

• 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. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv
• Amine, Oussama; Coffie, Emmanuel; Harang, Fabian Andsem & Proske, Frank Norbert (2020). Bismut–Elworthy–Li formula, singular SDEs, fractional Brownian motion, Malliavin calculus, stochastic flows, stochastic volatility. Communications in Mathematical Sciences. ISSN 1539-6746. 18(7), s. 1863–1890. doi: 10.4310/CMS.2020.v18.n7.a3. Fulltekst i vitenarkiv
• Baños, David; Bauer, Martin; Meyer-Brandis, Thilo & Proske, Frank Norbert (2019). Restoration of Well-Posedness of Infinite-dimensional Singular ODE's via Noise. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv
• Baños, David Ruiz; Nilssen, Torstein Kastberg & Proske, Frank Norbert (2019). Strong Existence and Higher Order Fréchet Differentiability of Stochastic Flows of Fractional Brownian Motion Driven SDEs with Singular Drift. Journal of Dynamics and Differential Equations. ISSN 1040-7294. 32, s. 1819–1866. doi: 10.1007/s10884-019-09789-4. Fulltekst i vitenarkiv Vis sammendrag

  In this paper we present a new method for the construction of strong solutions of SDE’s with merely integrable drift coefficients driven by a multidimensional fractional Brownian motion with Hurst parameter H<1/2. Furthermore, we prove the rather surprising result of the higher order Fréchet differentiability of stochastic flows of such SDE’s in the case of a small Hurst parameter. In establishing these results we use techniques from Malliavin calculus combined with new ideas based on a “local time variational calculus”. We expect that our general approach can be also applied to the study of certain types of stochastic partial differential equations as e.g. stochastic conservation laws driven by rough paths.

• Agram, Nacira; Bachouch, Achref; Øksendal, Bernt & Proske, Frank Norbert (2019). Singular control and optimal stopping of memory mean-field processes. SIAM Journal on Mathematical Analysis. ISSN 0036-1410. 51(1), s. 450–468. doi: 10.1137/18M1174787. Fulltekst i vitenarkiv Vis sammendrag

  The purpose of this paper is to study the following topics and the relation between them: (i) Optimal singular control of mean-field stochastic differential equations with memory; (ii) reflected advanced mean-field backward stochastic differential equations; and (iii) optimal stopping of mean-field stochastic differential equations. More specifically, we do the following: (1) We prove the existence and uniqueness of the solutions of some reflected advanced memory backward stochastic differential equations; (2) we give sufficient and necessary conditions for an optimal singular control of a memory mean-field stochastic differential equation (MMSDE) with partial information; and (3) we deduce a relation between the optimal singular control of an MMSDE and the optimal stopping of such processes.

• Banos, David; Bølviken, Erik; Duedahl, Sindre & Proske, Frank Norbert (2019). Modeling and estimation of stochastic transition rates in life insurance with regime switching based on generalized Cox processes. Scandinavian Actuarial Journal. ISSN 0346-1238. doi: 10.1080/03461238.2019.1636858. Fulltekst i vitenarkiv Vis sammendrag

  In this paper we aim at modeling stochastic transition rates of state processes in life insurance by using generalized Cox processes. A feature of a our non-Gaussian model is that it can be used to capture ‘regime switching’ effects of data which may be due to regulatory changes in insurance markets or external ‘shocks’ caused e.g. by an economical crisis, natural disasters or epidemics. We propose a method how to estimate the unknown parameters of our model for stochastic transition rates from insurance data by using non-linear filtering techniques for Lévy processes. As a result we also obtain an explicit formula for the unnormalized density of a filtering problem with singular coefficients.

• Baños, David Ruiz; Duedahl, Sindre; Meyer-Brandis, Thilo & Proske, Frank Norbert (2018). Construction of Malliavin differentiable strong solutions of SDEs under an integrability condition on the drift without the Yamada-Watanabe principle. Annales de l'I.H.P. Probabilites et statistiques. ISSN 0246-0203. 54(3), s. 1464–1491. doi: 10.1214/17-AIHP845. Fulltekst i vitenarkiv Vis sammendrag

  In this paper we aim at employing a compactness criterion of Da Prato, Malliavin, Nualart (C. R. Math. Acad. Sci. Paris 315 (1992) 1287–1291) for square integrable Brownian functionals to construct strong solutions of SDE’s under an integrability condition on the drift coefficient. The obtained solutions turn out to be Malliavin differentiable and are used to derive a Bismut–Elworthy–Li formula for solutions of the Kolmogorov equation. We emphasise that our approach exhibits high flexibility to study a variety of other types of stochastic (partial) differential equations as e.g. stochastic differential equations driven by fractional Brownian motion.

• Baños, David; Nunno, Giulia Di; Haferkorn, Hannes Hagen & Proske, Frank Norbert (2018). Stochastic functional differential equations and sensitivity to their initial path. I Celledoni, Elena; Nunno, Giulia Di; Ebrahimi-Fard, Kurusch & Munthe-Kaas, Hans (Red.), Computation and Combinatorics in Dynamics, Stochastics and Control. Springer. ISSN 9783030015923. s. 37–70. doi: 10.1007/978-3-030-01593-0_2. Fulltekst i vitenarkiv Vis sammendrag

  We consider systems with memory represented by stochastic functional differential equations. Substantially, these are stochastic differential equations with coefficients depending on the past history of the process itself. Such coefficients are hence defined on a functional space. Models with memory appear in many applications ranging from biology to finance. Here we consider the results of some evaluations based on these models (e.g. the prices of some financial products) and the risks connected to the choice of these models. In particular we focus on the impact of the initial condition on the evaluations. This problem is known as the analysis of sensitivity to the initial condition and, in the terminology of finance, it is referred to as the Delta. In this work the initial condition is represented by the relevant past history of the stochastic functional differential equation. This naturally leads to the redesign of the definition of Delta. We suggest to define it as a functional directional derivative, this is a natural choice. For this we study a representation formula which allows for its computation without requiring that the evaluation functional is differentiable. This feature is particularly relevant for applications. Our formula is achieved by studying an appropriate relationship between Malliavin derivative and functional directional derivative. For this we introduce the technique of randomisation of the initial condition.

• Agram, Nacira; Haadem, Sven; Øksendal, Bernt & Proske, Frank Norbert (2018). Optimal stopping, randomized stopping and singular control with partial information flow. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv
• Bauer, Martin; Meyer-Brandis, Thilo & Proske, Frank Norbert (2018). Strong solutions of mean-field stochastic differential equations with irregular drift. Electronic Journal of Probability (EJP). 23(132), s. 1–35. doi: 10.1214/18-EJP259. Fulltekst i vitenarkiv Vis sammendrag

  We investigate existence and uniqueness of strong solutions of mean-field stochastic differential equations with irregular drift coefficients. Our direct construction of strong solutions is mainly based on a compactness criterion employing Malliavin Calculus together with some local time calculus. Furthermore, we establish regularity properties of the solutions such as Malliavin differentiablility as well as Sobolev differentiability and Hölder continuity in the initial condition. Using this properties we formulate an extension of the Bismut-Elworthy-Li formula to mean-field stochastic differential equations to get a probabilistic representation of the first order derivative of an expectation functional with respect to the initial condition.

• Amine, Oussama; Banos, David & Proske, Frank Norbert (2018). Regularity Properties of the Stochastic Flow of a Skew Fractional Brownian Motion. arXiv.org. ISSN 2331-8422. doi: 10.1142/s0219025720500058. Fulltekst i vitenarkiv
• Amine, Oussama; Coffie, Emmanuel; Harang, Fabian Andsem & Proske, Frank Norbert (2018). A Bismut-Elworthy-Li Formula for Singular SDE's Driven by a Fractional Brownian Motion and Applications to Rough Volatility Modeling. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv
• Pilipenko, Andrey & Proske, Frank Norbert (2018). On perturbations of an ODE with non-Lipschitz coefficients by a small self-similar noise. Statistics and Probability Letters. ISSN 0167-7152. 132, s. 62–73. doi: 10.1016/j.spl.2017.09.005. Fulltekst i vitenarkiv
• Pilipenko, Andrey & Proske, Frank Norbert (2018). On a selection problem for small noise perturbation in the multidimensional case. Stochastics and Dynamics. ISSN 0219-4937. 18(6). doi: 10.1142/S0219493718500454. Fulltekst i vitenarkiv Vis sammendrag

  The problem on identification of a limit of an ordinary differential equation with discontinuous drift that perturbed by a zero-noise is considered in multidimensional case. This problem is a classical subject of stochastic analysis, see, for example, [6, 29, 11, 20]. However the multidimensional case was poorly investigated. We assume that the drift coefficient has a jump discontinuity along a hyperplane and is Lipschitz continuous in the upper and lower half-spaces. It appears that the behavior of the limit process depends on signs of the normal component of the drift at the upper and lower half-spaces in a neighborhood of the hyperplane, all cases are considered. Electronic version of an article published as Pilipenko, Andrey, and Frank Norbert Proske. "On a selection problem for small noise perturbation in the multidimensional case." Stochastics and Dynamics (2018): 1850045. https://doi.org/10.1142/S0219493718500454 © World Scientific Publishing Company https://www.worldscientific.com/worldscinet/sd

• Baños, David Ruiz; Meyer-Brandis, Thilo; Proske, Frank Norbert & Duedahl, Sindre (2017). Computing Deltas without Derivatives. Finance and Stochastics. ISSN 0949-2984. 21(2), s. 509–549. doi: 10.1007/s00780-016-0321-3. Fulltekst i vitenarkiv Vis sammendrag

  A well-known application of Malliavin calculus in mathematical finance is the probabilistic representation of option price sensitivities, the so-called Greeks, as expectation functionals that do not involve the derivative of the payoff function. This allows numerically tractable computation of the Greeks even for discontinuous payoff functions. However, while the payoff function is allowed to be irregular, the coefficients of the underlying diffusion are required to be smooth in the existing literature, which for example already excludes simple regime-switching diffusion models. The aim of this article is to generalise this application of Malliavin calculus to Itô diffusions with irregular drift coefficients, where we focus here on the computation of the delta, which is the option price sensitivity with respect to the initial value of the underlying. To this end, we first show existence, Malliavin differentiability and (Sobolev) differentiability in the initial condition for strong solutions of Itô diffusions with drift coefficients that can be decomposed into the sum of a bounded, but merely measurable, and a Lipschitz part. Furthermore, we give explicit expressions for the corresponding Malliavin and Sobolev derivatives in terms of the local time of the diffusion, respectively. We then turn to the main objective of this article and analyse the existence and probabilistic representation of the corresponding deltas for European and path-dependent options. We conclude with a small simulation study of several regime-switching examples. The final version of this research has been published by Finance and Stochastics. © Springer Verlag.

• Baños, David Ruiz; Haferkorn, Hannes Hagen & Proske, Frank Norbert (2017). Strong Uniqueness of Singular Stochastic Delay Equations. arXiv.org. ISSN 2331-8422. Fulltekst i vitenarkiv
• Pilipenko, Andrey & Proske, Frank Norbert (2016). On perturbations of an ODE with non-Lipschitz coefficients by a small self-similar noise. arXiv.org. ISSN 2331-8422. doi: 10.1016/j.spl.2017.09.005. Fulltekst i vitenarkiv
• Mohammed, Salah-Eldin; Nilssen, Torstein Kastberg & Proske, Frank Norbert (2015). Sobolev differentiable stochastic flows for sdes with singular coefficients: Applications to the transport equation. Annals of Probability. ISSN 0091-1798. 43(3), s. 1535–1576. doi: 10.1214/14-AOP909. Fulltekst i vitenarkiv
• Pilipenko, Andrey & Proske, Frank Norbert (2015). On a Selection Problem for Small Noise Perturbation in Multidimensional Case. arXiv.org. ISSN 2331-8422. doi: 10.1142/s0219493718500454. Fulltekst i vitenarkiv
• Banos, David Ruiz; Nilssen, Torstein Kastberg & Proske, Frank Norbert (2015). Strong existence and higher order Fréchet differentiability of stochastic flows of fractional Brownian motion driven SDE's with singular drift. arXiv.org. ISSN 2331-8422. doi: 10.1007/s10884-019-09789-4. Fulltekst i vitenarkiv
• Bølviken, Erik; Proske, Frank Norbert & Rubtsov, Mark (2014). Pricing of Margrabe options for large investors with application to asset-liability management in life insurance. Journal of Mathematical Finance. ISSN 2162-2434. 4(2), s. 113–122. doi: 10.4236/jmf.2014.42011. Fulltekst i vitenarkiv
• Haadem, Sven & Proske, Frank Norbert (2014). On the Construction and Malliavin Differentiability of Solutions of Levy Noise driven SDE's with Singular Coefficients. Journal of Functional Analysis. ISSN 0022-1236. 266(8), s. 5321–5359. doi: 10.1016/j.jfa.2014.02.009. Fulltekst i vitenarkiv
• Pamen, Olivier Menoukeu; Proske, Frank Norbert & Salleh, Hassilah Binti (2014). Stochastic differential games in insider markets via Malliavin calculus. Journal of Optimization Theory and Applications. ISSN 0022-3239. 160(1), s. 302–343. doi: 10.1007/s10957-013-0310-z. Fulltekst i vitenarkiv
• Agram, Nacira; Haadem, Sven; Øksendal, Bernt & Proske, Frank Norbert (2013). A maximum principle for infinite horizon delay equations. SIAM Journal on Mathematical Analysis. ISSN 0036-1410. 45, s. 2499–2522. doi: 10.1137/120882809. Fulltekst i vitenarkiv
• Kettler, Paul Carlisle; Yablonski, Aleh & Proske, Frank Norbert (2013). Market microstructure and price discovery. Journal of Mathematical Finance. ISSN 2162-2434. 3(1). doi: 10.4236/jmf.2013.31001. Fulltekst i vitenarkiv
• Menoukeu, Olivier Pamen; Meyer-Brandis, Thilo; Nilssen, Torstein Kastberg; Proske, Frank Norbert & Zhang, Tusheng (2013). A variational approach to the construction and Malliavin differentiability of strong solutions of SDE's. Mathematische Annalen. ISSN 0025-5831. 357(2), s. 761–799. doi: 10.1007/s00208-013-0916-3. Fulltekst i vitenarkiv
• Pamen, Olivier Menoukeu; Meyer-Brandis, Thilo; Proske, Frank Norbert & Salleh, Hassilah Binti (2013). Malliavin Calculus Applied to Optimal Control of Stochastic Partial Differential Equations with Jumps. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. 85(3), s. 431–463. doi: 10.1080/17442508.2011.652964. Fulltekst i vitenarkiv
• Agram, Nacira; Haadem, Sven; Øksendal, Bernt & Proske, Frank Norbert (2013). A maximum principle for infinite horizon delay equatations. SIAM Journal on Mathematical Analysis. ISSN 0036-1410. 45(4), s. 2499–2522. doi: 10.1137/120882809. Fulltekst i vitenarkiv Vis sammendrag

  We prove a maximum principle of optimal control of stochastic delay equations on infinite horizon. We establish first and second sufficient stochastic maximum principles as well as necessary conditions for that problem. We illustrate our results with an application to the optimal consumption rate from an economic quantity.

• Haadem, Sven; Øksendal, Bernt & Proske, Frank Norbert (2013). Maximum principles for jump diffusion processes with infinite horizon. Automatica. ISSN 0005-1098. 49(7), s. 2267–2275. doi: 10.1016/j.automatica.2013.04.011. Fulltekst i vitenarkiv Vis sammendrag

  We prove maximum principles for the problem of optimal control for a jump diffusion with infinite horizon and partial information. The results are applied to an optimal consumption and portfolio problem in infinite horizon. NOTICE: this is the author’s version of a work that was accepted for publication in Automatica. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Automatica

• Yolcu, Okur Yeliz; Proske, Frank Norbert & Salleh, Hassilah Binti (2012). SDE Solutions in the Space of Smooth Random Variables. Communications on Stochastic Analysis. ISSN 0973-9599. 6(3), s. 451–470. doi: 10.31390/cosa.6.3.07. Fulltekst i vitenarkiv
• Proske, Frank Norbert; Rubtsov, Mark & Ta, An Thi Kieu (2011). RISK INDIFFERENCE PRICING OF FUNCTIONAL CLAIMS OF THE YIELD SURFACE IN THE PRESENCE OF PARTIAL INFORMATION TA. Communications on Stochastic Analysis. ISSN 0973-9599. 5(3), s. 541–563. doi: 10.31390/cosa.5.3.06. Fulltekst i vitenarkiv
• Mandrekar, Vidyadhar; Meyer-Brandis, Thilo & Proske, Frank Norbert (2011). A Bayes Formula for Nonlinear Filtering with Gaussian and Cox Noise. Journal of Probability and Statistics. ISSN 1687-952X. doi: 10.1155/2011/259091. Fulltekst i vitenarkiv
• Nunno, Giulia Di; Pamen, Olivier Menoukeu; Øksendal, Bernt & Proske, Frank Norbert (2011). A general maximum principle for anticipative stochastic control and applications to insider trading. I Nunno, Giulia Di & Øksendal, Bernt (Red.), Advanced Mathematical Methods for Finance. Springer. ISSN 9783642184116. s. 181–221. doi: 10.1007/978-3-642-18412-3_7. Fulltekst i vitenarkiv
• Kettler, Paul Carlisle; Proske, Frank Norbert & Rubtsov, Mark (2011). Sensitivity with respect to the yield curve: duration in a stochastic setting. Springer Series in Statistics. ISSN 0172-7397. s. 363–385. doi: 10.1007/978-3-319-02069-3_17. Fulltekst i vitenarkiv
• Nunno, Giulia Di; Øksendal, Bernt; Pamen, Olivier Menoukeu & Proske, Frank Norbert (2011). Uniqueness of Decompositions of Skorohod-Semimartingales. Infinite Dimensional Analysis Quantum Probability and Related Topics. ISSN 0219-0257. 14(1), s. 15–24. doi: 10.1142/S0219025711004274. Fulltekst i vitenarkiv
• Ta, An Thi Kieu; Proske, Frank Norbert & Rubtsov, Mark (2010). A SPDE maximum principle for stochastic differential games under partial information with application to optimal portfolios on fixed income markets. Stochastics: An International Journal of Probability and Stochastic Processes. ISSN 1744-2508. 82(1), s. 3–23. doi: 10.1080/17442500902723542. Fulltekst i vitenarkiv
• Meyer-Brandis, Thilo & Proske, Frank Norbert (2010). Explicit Representation of Strong Solutions of SDEs Driven by Infinite-Dimensional L,vy Processes. Journal of theoretical probability. ISSN 0894-9840. 23(1), s. 301–314. doi: 10.1007/s10959-009-0226-6. Fulltekst i vitenarkiv
• Meyer-Brandis, Thilo & Proske, Frank Norbert (2010). Construction of strong solutions of SDE's via Malliavin calculus. Journal of Functional Analysis. ISSN 0022-1236. 258(11), s. 3922–3953. doi: 10.1016/j.jfa.2009.11.010. Fulltekst i vitenarkiv

Se alle arbeider i NVA

• 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
• 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
• Proske, Frank Norbert (2019). Strong solutions of SDE's with generalized drift and multidimensional fractional Brownian initial nise. Fulltekst i vitenarkiv
• Baños, David Ruiz; Ortiz-Latorre, Salvador; Pilipenko, Andrey & Proske, Frank Norbert (2017). Strong solutions of SDE's with generalized drift and multidimensional fractional Brownian initial noise. Fulltekst i vitenarkiv
• Nilssen, Torstein Kastberg; Baños, David Ruiz & Proske, Frank Norbert (2016). Strong Existence and higher order differentiability of stochastic flows of fractional Brownian motion driven SDE's with singular drift. Fulltekst i vitenarkiv
• Proske, Frank Norbert & Haadem, Sven (2014). On the construction and Malliavin differentiability of solutions of Lévy noise driven SDE's with singular coefficients. Fulltekst i vitenarkiv
• Haadem, Sven & Proske, Frank Norbert (2013). On the Construction and Malliavin Differentiability of Strong Solutions to Levy Noise Driven SDEs with Singular Coefficients. Fulltekst i vitenarkiv
• Proske, Frank Norbert (2012). Construction and Malliavin differentiability of strong solutions of SDEs with merely measurable drift. Fulltekst i vitenarkiv
• Nunno, Giulia Di; L'Aurora, Edoardo; Moschetta, Marina; Proske, Frank Norbert & Ruiz-Banos, David (2012). Market with memory: pricing and sensitivity analysis. Fulltekst i vitenarkiv
• Nunno, Giulia Di; L'Aurora, Edoardo; Moschetta, Marina; Proske, Frank Norbert & Ruiz-Banos, David (2012). Market with memory and sensitivity to the past. Fulltekst i vitenarkiv
• Proske, Frank Norbert (2010). Computing Greeks without Derivatives (joint paper with Menoukeu-Pamen, O., Meyer-Brandis, T. in progress). Fulltekst i vitenarkiv
• Baños, David; Nunno, Giulia Di & Proske, Frank Norbert (2013). SENSITIVITY ANALYSIS IN A MARKET WITH MEMORY. Universitetet i Oslo. Fulltekst i vitenarkiv Vis sammendrag

  A general market model with memory is considered. The formulation is given in terms of stochastic functional di erential equations, which allow for exibility in the modeling of market memory and delays. We focus on the sensitivity analysis of the dependence of option prices on the memory. This implies a generalization of the concept of delta. Our techniques use Malliavin calculus and Fréchet derivation. When it comes to option prices, we consider both the risk-neutral and the benchmark approaches and we compute the delta in both cases. Some examples are provided.

• Flandoli, Franco; Nilssen, Torstein Kastberg & Proske, Frank Norbert (2013). Malliavin differentiability and strong solutions for a class of SDE in Hilbert spaces. Universitetet i Oslo. Fulltekst i vitenarkiv Vis sammendrag

  We consider a class of Hilbert-space valued SDE’s where the drift coefficients are non- Lipschitzian in the sense of Hölder-continuity. Using a novel technique based on Malliavin calculus we show in this paper the existence and uniqueness of a mild solution to such equations. We emphasize that our approach does not rely on the Yamada-Watanabe principle. Moreover our method gives the important additional insight that the obtained solution is Malliavin differentiable - a property which was recently shown to play a crucial role in the study of the geometry of certain optimal causal transference plans, [12].

• Nilssen, Torstein Kastberg; Mohammed, Salah-Eldin & Proske, Frank Norbert (2012). Sobolev Differentiable Stochastic Flows for SDE’s with SingularCoefficients: Applications to the Transport Equation. Universitetet i Oslo. Fulltekst i vitenarkiv
• Haadem, Sven; Øksendal, Bernt & Proske, Frank Norbert (2012). Maximum principles for jump diffusion processes with infinite horizon. Universitetet i Oslo. doi: 10.1016/j.automatica.2013.04.011. Fulltekst i vitenarkiv Vis sammendrag

  We prove maximum principles for the problem of optimal control for a jump diffusion with infinite horizon and partial information. The results are applied to partial information optimal consumption and portfolio problems in infinite horizon.

• Agram, Nacira; Haadem, Sven; Øksendal, Bernt & Proske, Frank Norbert (2012). A MAXIMUM PRINCIPLE FOR INFINITE HORIZON DELAY EQUATIONS. Universitetet i Oslo. Fulltekst i vitenarkiv Vis sammendrag

  We prove a maximum principle of optimal control of stochastic delay equations on infinite horizon. We establish first and second sufficient stochastic maximum principles as well as necessary conditions for that problem. We illustrate our results by an application to the optimal consumption rate from an economic quantity.

• Pamen, Olivier Menoukeu; Meyer-Brandis, Thilo; Nilssen, Torstein Kastberg; Proske, Frank Norbert & Zhang, Tusheng (2011). A variational approach to the construction and Malliavin differentiability of strong solutions of SDE's. Universitetet i Oslo. Fulltekst i vitenarkiv
• Proske, Frank Norbert; Rubtsov, Mark & Ta, An Thi Kieu (2010). Risk Indifference Pricing of Functional Claims of the Yield Surface in the Presence of Partial Information. Universitetet i Oslo. Fulltekst i vitenarkiv Vis sammendrag

  In this paper we study the problem of risk indifference pricing of interest rate claims which are functionals of a bond yield surface under partial information. Our approach to solve this problem relies on a maximum principle for partial information control of stochastic differential games based on generalized bond portfolios. The latter method enables us to establish an explicit representation of the risk indifference price of such claims.

• Meyer-Brandis, Thilo; Proske, Frank Norbert & Pamen, Olivier Menoukeu (2010). A GEL'FAND TRIPLE APPROACH TO THE SMALL NOISE PROBLEM FOR DISCONTINUOUS ODE'S. Universitetet i Oslo. Fulltekst i vitenarkiv Vis sammendrag

  In this paper, we develop a variational approach to study perturbation problems of ordinary differential equations (ODE's) with discontinuous coefficients. We propose a mathematical framework which can be used to construct stable (and regular) solution processes of discontinuous ODE's.

• Kettler, Paul Carlisle; Proske, Frank Norbert & Rubtsov, Mark (2010). SENSITIVITY WITH RESPECT TO THE YIELD CURVE: DURATION IN A STOCHASTIC SETTING. Universitetet i Oslo. Fulltekst i vitenarkiv Vis sammendrag

  Bond duration in its basic deterministic form is a concept well understood. Its meaning in the context of a yield curve on a stochastic path is less well developed. We extend the basic idea to a stochastic setting. More precisely, we introduce the concept of stochastic duration as a Malliavin derivative in the direction of a stochastic yield surface modeled by the Musiela equation. Further, using this concept we also propose a mathematical framework for the construction of immunization strategies (or delta hedges) of portfolios of interest-rate-sensitive securities with respect to the fluctuation of the whole yield surface.

• Yolcu, Okur Yeliz; Proske, Frank Norbert & Salleh, Hassilah Binti (2009). SDE SOLUTIONS IN THE SPACE OF SMOOTH RANDOM VARIABLES. Universitetet i Oslo. Fulltekst i vitenarkiv
• Pamen, Olivier Menoukeu; Proske, Frank Norbert & Salleh, Hassilah Binti (2009). Stochastic Differential Games in Insider Markets via Malliavin Calculus. Universitetet i Oslo. Fulltekst i vitenarkiv Vis sammendrag

  In this paper we use techniques of Malliavin calculus and forward integration to present a general stochastic maximum principle for anticipating stochastic differential equations driven by a Lévy type of noise. We apply our result to study a general stochastic differential game problem of an insider.

• Nunno, Giulia Di; Øksendal, Bernt Karsten; Pamen, Olivier Menoukeu & Proske, Frank Norbert (2009). A general maximum principle for anticipative stochastic control and applications to insider trading. Universitetet i Oslo. Fulltekst i vitenarkiv Vis sammendrag

  In this paper we suggest a general stochastic maximum principle for optimal control of anticipating stochastic differential equations driven by a Lévy type of noise. We use techniques of Malliavin calculus and forward integration. We apply our results to study a general optimal portfolio problem of an insider. In particular, we find conditions on the insider information filtration which are sufficient to give the insider an infinite wealth. We also apply the results to find the optimal consumption rate for an insider.

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