Asymptotic Expansion for Forward-Backward SDEs with Jumps

TL;DR

This paper introduces an asymptotic expansion method for approximating decoupled forward-backward stochastic differential equations with jumps, providing a semi-analytic solution with error estimates, applicable to various jump processes.

Contribution

It develops a novel asymptotic expansion technique for FBSDEs with jumps driven by Poisson measures and Brownian motion, including error analysis and applicability to non-Poissonian jumps.

Findings

Provides a semi-analytic solution method solving linear ODEs.

Handles state-dependent and non-Poissonian jumps.

Offers error estimates for the approximation.

Abstract

This work provides a semi-analytic approximation method for decoupled forwardbackward SDEs (FBSDEs) with jumps. In particular, we construct an asymptotic expansion method for FBSDEs driven by the random Poisson measures with {\sigma}-finite compensators as well as the standard Brownian motions around the small-variance limit of the forward SDE. We provide a semi-analytic solution technique as well as its error estimate for which we only need to solve essentially a system of linear ODEs. In the case of a finite jump measure with a bounded intensity, the method can also handle state-dependent and hence non-Poissonian jumps, which are quite relevant for many practical applications.