Giulia Di Nunno

• 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.

• 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.

• 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
• 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.

• 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.

• 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
• 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
• 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
• 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
• 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.

• 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
• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• Benth, Fred Espen; Nunno, Giulia Di & Schroers, Dennis (2021). Copula measures and Sklar's theorem in arbitrary dimensions. Scandinavian Journal of Statistics. ISSN 0303-6898. doi: 10.1111/sjos.12559. Fulltekst i vitenarkiv
• Benth, Fred Espen; Nunno, Giulia Di & Simonsen, Iben Cathrine (2021). Sensitivity analysis in the infinite dimensional Heston model. Infinite Dimensional Analysis Quantum Probability and Related Topics. ISSN 0219-0257. 24(2). doi: 10.1142/S0219025721500144. Fulltekst i vitenarkiv

Se alle arbeider i NVA

• 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.

Se alle arbeider i NVA

• 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
• 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
• Nunno, Giulia Di (2025). Utility maximisation and change of variable formulas for time-changed dynamics. 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
• 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
• Nunno, Giulia Di (2024). Deep Operator BSDE. 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
• 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
• 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
• 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
• 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
• 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
• 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.

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