A Discretization Scheme for BSDEs with Random Time Horizon

TL;DR

This paper extends the backward Euler method to BSDEs with random, potentially unbounded time horizons, providing strong error bounds and applications to decoupled FBSDEs with exit time approximations.

Contribution

It introduces a discretization scheme for BSDEs with random time horizons and derives explicit error bounds considering various approximation factors.

Findings

Error bounds depend on stepsize, terminal time approximation, and generator approximation.

The scheme applies to decoupled FBSDEs on bounded domains.

Refined bounds depend only on exit time approximation.

Abstract

We analyze a natural extension of the backward Euler approximation for a class of BSDEs with Lipschitz generators and random (unbounded) time horizons. We derive strong error bounds in terms of the underlying stepsize; the distance between the continuous terminal time and a discrete-time approximation; the distance between the terminal condition and a respective approximation; and an integrated distance depending on an approximation of the time component of the generator - all are scaled by the exponential of the maximal terminal time. As application we consider decoupled FBSDEs on bounded domains. We use an Euler-Maruyama scheme to approximate the diffusion and further refine our error bounds to only depend on the distance of the exit times.