A deep solver for BSDEs with jumps

TL;DR

This paper extends a deep learning-based solver for stochastic differential equations to handle equations with jumps, enabling efficient high-dimensional option pricing and credit risk applications.

Contribution

It introduces a neural network-based method for solving FBSDEs with jumps, including infinite activity cases, advancing numerical solutions for complex stochastic models.

Findings

Successfully applied to high-dimensional option pricing

Handles both finite and infinite jump activity

Demonstrates applicability to counterparty credit risk

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.