A decomposition approach for the discrete-time approximation of FBSDEs with a jump

TL;DR

This paper introduces a recursive discretization scheme for solving Forward-Backward Stochastic Differential Equations with jumps, demonstrating convergence and error bounds for both Lipschitz and quadratic growth generators.

Contribution

It extends existing discretization methods to FBSDEs with jumps, providing convergence analysis and rates comparable to Brownian FBSDE schemes.

Findings

The proposed scheme converges as the number of steps increases.

Error bounds are established for Lipschitz and quadratic growth generators.

The convergence rate matches that of schemes for Brownian FBSDEs.

Abstract

We are concerned with the discretization of a solution of a Forward-Backward stochastic differential equation (FBSDE) with a jump process depending on the Brownian motion. In this paper, we study the cases of Lipschitz generators and the generators with a quadratic growth w.r.t. the variable z. We propose a recursive scheme based on a general existence result given in a companion paper and we study the error induced by the time discretization. We prove the convergence of the scheme when the number of time steps n goes to infinity. Our approach allows to get a convergence rate similar to that of schemes of Brownian FBSDEs.