A deep solver for BSDEs with jumps

TL;DR

This work extends the Deep BSDE solver to forward–backward SDEs with jumps, enabling efficient numerical treatment of high-dimensional jump-diffusion models via time discretization and neural network parameterization of controls. It introduces dual neural-network structures to approximate both value/gradient terms and jump-related integrals, and augments the loss with a martingale-structure penalty to enforce jump contributions accurately; for infinite activity, it combines small-jump diffusion approximations with a compound Poisson large-jump representation and provides an a priori error analysis for the forward and backward components. The method is validated through diverse financial examples, including pure-jump, single-asset and basket options up to dimension 100, CGMY (infinite activity) models, and an application to counterparty credit risk (CVA), demonstrating feasibility and scalability in high dimensions. Collectively, the paper contributes a practical framework for solving high-dimensional PIDEs via FBSDEs with jumps and lays groundwork for future convergence theory and risk-management applications in jump models.

Abstract

The aim of this work is to propose an extension of the deep solver by Han, Jentzen, E (2018) to the case of forward backward stochastic differential equations (FBSDEs) with jumps. As in the aforementioned solver, starting from a discretized version of the FBSDE and parametrizing the (high dimensional) control processes by means of a family of artificial neural networks (ANNs), the FBSDE is viewed as a model-based reinforcement learning problem and the ANN parameters are fitted so as to minimize a prescribed loss function. We take into account both finite and infinite jump activity by introducing, in the latter case, an approximation with finitely many jumps of the forward process. We successfully apply our algorithm to option pricing problems in low and high dimension and discuss the applicability in the context of counterparty credit risk.

Figures (7)

• Figure 1: Left: Approximate initial value of the BSDE against the number of training steps (the number of batch iterations). Right: 20 representative paths of our approximations compared to the analytical solutions. Note that only four paths contain at least one jump.
• Figure 2: Left: Approximate initial value of the BSDE against the number of training steps (the number of batch iterations). Right: Five representative paths of our approximations compared to the analytical solutions.
• Figure 3: Left: Approximate initial value of the BSDE for $d=5$ (upper), $d=25$ (middle) and $d=100$ (lower) against the number of training steps (the number of batch iterations). Right: Five representative paths of our approximations compared to reference solutions obtained from a nested Monte Carlo approximation for $d=5$ (upper), $d=25$ (middle) and $d=100$ (lower).
• Figure 4: Comparison of log-price densities of a single asset for different values of $\lambda$.
• Figure 5: Histogram of empirical density in orange and true density in blue.

Theorems & Definitions (15)

• Theorem 2.1
• proof
• Remark 2.2
• Remark 2.3
• Theorem 2.4
• Theorem 2.5: Theorem 4.2.1, Delongbook
• Lemma 2.6
• proof
• Remark 3.1
• Remark 3.2