A forward scheme with machine learning for forward-backward SDEs with jumps by decoupling jumps

TL;DR

This paper introduces a machine learning-based forward scheme for high-dimensional FBSDEs with jumps, which decouples jumps at each step, converges rapidly, and effectively approximates solutions without directly handling jumps.

Contribution

It develops a novel recursive forward scheme that decouples jumps in high-dimensional FBSDEs using neural networks, with proven exponential convergence and broad applicability.

Findings

Scheme accurately approximates high-dimensional FBSDEs with jumps up to 100 dimensions.

Convergence is exponential and requires few iterations for sufficient accuracy.

Neural network implementation enables handling of jumps without explicit jump simulation.

Abstract

Forward-backward stochastic differential equations (FBSDEs) have been generalized by introducing jumps for better capturing random phenomena, while the resulting FBSDEs are far more intricate than the standard one from every perspective. In this work, we establish a forward scheme for potentially high-dimensional FBSDEs with jumps, taking a similar approach to [Bender and Denk, 117 (2007), Stoch. Process. Their Appl., pp.1793-1812], with the aid of machine learning techniques for implementation. The developed forward scheme is built upon a recursive representation that decouples random jumps at every step and converges exponentially fast to the original FBSDE with jumps, often requiring only a few iterations to achieve sufficient accuracy, along with the error bound vanishing for lower jump intensities. The established framework also holds novelty in its neural network-based implementation of a wide class of forward schemes for FBSDEs, notably whether with or without jumps. We provide an extensive collection of numerical results, showcasing the effectiveness of the proposed recursion and its corresponding forward scheme in approximating high-dimensional FBSDEs with jumps (up to 100-dimension) without directly handling the random jumps.