A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications

TL;DR

This paper establishes a fundamental mean-square convergence theorem for SDEs with polynomially growing coefficients under a one-sided Lipschitz condition, and demonstrates its application to various numerical methods with supporting numerical tests.

Contribution

It introduces a new convergence theorem for SDEs with non-globally Lipschitz coefficients and applies it to analyze several numerical schemes.

Findings

The theorem guarantees mean-square convergence for specific SDEs.

Numerical tests confirm the theoretical convergence results.

Applications include balanced and implicit numerical methods.

Abstract

A version of the fundamental mean-square convergence theorem is proved for stochastic differential equations (SDE) which coefficients are allowed to grow polynomially at infinity and which satisfy a one-sided Lipschitz condition. The theorem is illustrated on a number of particular numerical methods, including a special balanced scheme and fully implicit methods. Some numerical tests are presented.