Numerical stability analysis of the Euler scheme for BSDEs

TL;DR

This paper investigates the numerical stability of the Euler scheme for solving Backward Stochastic Differential Equations (BSDEs), providing theoretical conditions and numerical illustrations to ensure stable approximations.

Contribution

It introduces a new notion of numerical stability for BSDE approximation schemes and establishes sufficient and necessary conditions for the Euler scheme's stability in various cases.

Findings

Sufficient conditions for Euler scheme stability in 1D and multidimensional cases.

Necessary conditions for stability in linear driver BSDEs.

Numerical examples validating the theoretical stability conditions.

Abstract

In this paper, we study the qualitative behaviour of approximation schemes for Backward Stochastic Differential Equations (BSDEs) by introducing a new notion of numerical stability. For the Euler scheme, we provide sufficient conditions in the one-dimensional and multidimensional case to guarantee the numerical stability. We then perform a classical Von Neumann stability analysis in the case of a linear driver $f$ and exhibit necessary conditions to get stability in this case. Finally, we illustrate our results with numerical applications.