Approximation and Error Analysis of Forward-Backward SDEs driven by General L\'evy Processes using Shot Noise Series Representations

TL;DR

This paper develops a method to simulate forward-backward stochastic differential equations driven by complex Lévy processes with infinite jump activity, using shot noise series to approximate the Lévy process and analyzing the associated errors.

Contribution

It introduces a novel approximation technique for Lévy-driven FBSDEs using shot noise series and provides error bounds for the approximation and discretization.

Findings

Derived $L^p$ error bounds for Lévy process approximation.

Established error estimates for the discretized FBSDE solutions.

Validated the approximation method under certain conditions.

Abstract

We consider the simulation of a system of decoupled forward-backward stochastic differential equations (FBSDEs) driven by a pure jump L\'evy process $L$ and an independent Brownian motion $B$. We allow the L\'evy process $L$ to have an infinite jump activity. Therefore, it is necessary for the simulation to employ a finite approximation of its L\'evy measure. We use the generalized shot noise series representation method by Rosinski (2001) to approximate the driving L\'evy process $L$. We compute the $L^p$ error, $p\ge2$, between the true and the approximated FBSDEs which arises from the finite truncation of the shot noise series (given sufficient conditions for existence and uniqueness of the FBSDE). We also derive the $L^p$ error between the true solution and the discretization of the approximated FBSDE using an appropriate backward Euler scheme.