Strong convergence rates for long-time approximations of SDEs with non-globally Lipschitz continuous coefficients

TL;DR

This paper establishes strong convergence rates for long-time numerical approximations of stochastic differential equations with non-globally Lipschitz coefficients, using backward Euler and projected Euler schemes, supported by numerical validation.

Contribution

It develops a long-time strong convergence theorem for general one-step schemes under non-globally Lipschitz conditions, and applies it to backward Euler and projected Euler methods.

Findings

Proves strong convergence rate over infinite time for specific schemes.

Validates theoretical results with numerical examples.

Abstract

This paper is concerned with long-time strong approximations of SDEs with non-globally Lipschitz coefficients.Under certain non-globally Lipschitz conditions, a long-time version of fundamental strong convergence theorem is established for general one-step time discretization schemes. With the aid of the fundamental strong convergence theorem, we prove the expected strong convergence rate over infinite time for two types of schemes such as the backward Euler method and the projected Euler method in non-globally Lipschitz settings. Numerical examples are finally reported to confirm our findings.