# A note on Asymptotic mean-square stability of stochastic linear two-step methods for SDEs

## TL;DR

This paper analyzes the asymptotic mean-square stability of various two-step stochastic numerical schemes for SDEs, providing conditions for stability and comparing their performance in scalar and multi-dimensional cases.

## Contribution

It derives necessary and sufficient stability conditions for two-step schemes applied to SDEs, including scalar and systems, and compares stability properties of different methods.

## Key findings

- BDF2 scheme is unconditionally stable for any step-size.
- AB2 and AM2 methods are unconditionally stable.
- Numerical experiments confirm theoretical stability conditions.

## Abstract

In this note we study the asymptotic mean-square stability for two-step schemes applied to a scalar stochastic differential equation (sde) and applied to systems of sdes. We derive necessary and sufficient conditions for the asymptotic MS-stability of the methods in terms of the parameters of the schemes. The stochastic Backward Differentiation Formula (BDF2) scheme is asymptotically mean-square stable for any step-size whereas the two-step Adams-Bashforth (AB2) and Adams-Moulton (AM2) methods are unconditionally stable. The improved versions of the schemes do not perform better w.r.t their stability behavior in the scalar case, as expected, but the situation is different in more dimensions. Numerical experiments confirm theoretical results.