Pere Diaz Lozano

• 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. Full text in Research Archive Show summary

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

• Lozano, Pere Diaz & Kloster, Thomas K. (2026). A Wiener Chaos Approach to Martingale Modelling and Implied Volatility Calibration. arXiv.org. ISSN 2331-8422. doi: 10.48550/ARXIV.2602.16232. Full text in Research Archive Show summary

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

• Diaz-Lozano, Pere & Nunno, Giulia Di (2026). Deep operator BSDE: A numerical scheme to approximate solution operators. The Annals of Applied Probability. ISSN 1050-5164. 36(3), p. 2594–2625. doi: 10.1214/25-aap2289. Full text in Research Archive Show summary

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

• Lozano, Pere Diaz & Nunno, Giulia Di (2025). An Euler scheme for BSDEs via the Wiener chaos decomposition. arXiv.org. ISSN 2331-8422. Full text in Research Archive Show summary

  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.

• Lozano, Pere Diaz; Bagén, Toni Lozano & Vives, Josep (2023). Neural stochastic differential equations for conditional time series generation using the Signature-Wasserstein-1 metric. Journal of Computational Finance. ISSN 1460-1559. 27(1), p. 1–23. doi: 10.21314/jcf.2023.005. Full text in Research Archive Show summary

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

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• Lozano, Pere Diaz (2026). Deep operator BSDE: A numerical scheme to approximate solution operators. Full text in Research Archive Show summary

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

• Lozano, Pere Diaz (2026). A Wiener Chaos Approach to Martingale Modelling and Implied Volatility Calibration. Full text in Research Archive Show summary

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

• Lozano, Pere Diaz (2026). A Wiener Chaos Approach to Martingale Modelling and Implied Volatility Calibration. Full text in Research Archive Show summary

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

• Lozano, Pere Diaz (2026). An Euler scheme for BSDEs via the Wiener chaos decomposition. Full text in Research Archive Show summary

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

• 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. Full text in Research Archive Show summary

  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.

• Lozano, Pere Diaz (2025). Deep operator BSDE: A numerical scheme to approximate solution operators. Full text in Research Archive Show summary

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

• Lozano, Pere Diaz (2023). Neural SDEs for Conditional Time Series Generation using the Signature Wasserstein-1 metric. Full text in Research Archive
• Lozano, Pere Diaz; Nunno, Giulia Di & Ortiz-Latorre, Salvador (2026). Backward stochastic equations and martingale modelling. Universitetet i Oslo. Full text in Research Archive Show summary

  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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