Convergence of a Deep BSDE solver with jumps

TL;DR

This work analyzes the convergence of a deep learning-based solver for forward-backward SDEs with jumps, translating the FBSDE/Feynman–Kac framework into a PIDE solution and providing both a posteriori and a priori error estimates for finite- and infinite-activity jump processes. It extends the jump-augmented deep BSDE solver to handle infinite activity via a regularization $X^{\\varepsilon}$ and leverages neural-network approximations of the control, with rigorous bounds that decompose errors into discretization, jump-approximation, and NN-approximation components. The results encompass contraction-based representations, path-regularity arguments, and Malliavin-calculus representations to relate surrogate quantities to the true $Z$ and $U$ processes, yielding practical guidance on grid size, jump-truncation level, and network accuracy. Overall, the paper offers a principled convergence theory for deep learning solvers of high-dimensional PIDEs arising from jump-diffusion models, with implications for stable and scalable numerical methods in finance and related fields.

Abstract

We study the error arising in the numerical approximation of FBSDEs and related PIDEs by means of a deep learning-based method. Our results focus on decoupled FBSDEs with jumps and extend the seminal work of HAn and Long (2020) analyzing the numerical error of the deep BSDE solver proposed in E et al. (2017). We provide a priori and a posteriori error estimates for the finite and infinite activity case.

Theorems & Definitions (31)

• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• Remark 3.2
• Remark 3.4
• Remark 3.5
• Lemma 3.6
• proof
• Theorem 3.7
• Lemma 3.8