Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients

TL;DR

This paper investigates the strong convergence and stability of Euler-Maruyama type numerical methods for stochastic differential equations with non-globally Lipschitz coefficients, relevant in finance and biology.

Contribution

It provides new results on the strong convergence and almost sure stability of implicit numerical schemes for nonlinear SDEs with non-Lipschitz coefficients.

Findings

Establishes strong convergence of EM schemes under non-Lipschitz conditions.

Proves almost sure asymptotic stability for the numerical methods.

Introduces a stochastic LaSalle principle for stability analysis.

Abstract

We are interested in the strong convergence and almost sure stability of Euler-Maruyama (EM) type approximations to the solutions of stochastic differential equations (SDEs) with non-linear and non-Lipschitzian coefficients. Motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence. In addition we examine the globally almost surely asymptotic stability in this non-linear setting for EM type schemes. In particular, we present a stochastic counterpart of the discrete LaSalle principle from which we deduce stability properties for numerical methods.