A decomposition approach for the discrete-time approximation of FBSDEs with a jump I : the Lipschitz case

TL;DR

This paper introduces a decomposition method for discretizing forward-backward stochastic differential equations with jumps in the Lipschitz case, leveraging recursive Brownian FBSDE systems for convergence analysis.

Contribution

It presents a novel approach linking FBSDEs with jumps to recursive Brownian FBSDEs, enabling effective discretization with proven convergence rates.

Findings

Achieves convergence rates comparable to Brownian FBSDE schemes

Links FBSDEs with jumps to recursive Brownian systems

Provides a practical discretization method for Lipschitz FBSDEs with jumps

Abstract

We study the discrete-time approximation for solutions of forward-backward stochas- tic dierential equations (FBSDEs) with a jump. In this part, we study the case of Lipschitz generators, and we refer to the second part of this work [15] for the quadratic case. Our method is based on a result given in the companion paper [14] which allows to link a FBSDE with a jump with a recursive system of Brownian FBSDEs. Then we use the classical results on discretization of Brownian FBSDEs to approximate the recursive system of FBSDEs and we recombine these approximations to get a dis- cretization of the FBSDE with a jump. This approach allows to get a convergence rate similar to that of schemes for Brownian FBSDEs.