Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients

TL;DR

This paper introduces a linear-theta-projected Euler scheme for semi-linear SDEs with non-globally Lipschitz coefficients, achieving exponential convergence to invariant measures and a weak convergence order of one, with computational advantages.

Contribution

The paper proposes a new explicit ergodicity-preserving scheme for semi-linear SDEs that handles non-globally Lipschitz coefficients and demonstrates its convergence properties.

Findings

The LTPE scheme converges exponentially to the invariant measure.

The weak error between numerical and true invariant measures is of order one.

The scheme has computational advantages over implicit methods.

Abstract

This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler (LTPE) scheme, which also admits an invariant measure, to handle the potential influence of the linear stiffness. Under certain assumptions, both the SDE and the corresponding LTPE method are shown to converge exponentially to the underlying invariant measures, respectively. Moreover, with time-independent regularity estimates for the corresponding Kolmogorov equation, the weak error between the numerical invariant measure and the original one can be guaranteed with convergence of order one. In terms of computational complexity, the proposed ergodicity preserving scheme with the nonlinearity explicitly treated has a significant advantage over the ergodicity preserving implicit Euler method in the literature. Numerical experiments are provided to verify our theoretical findings.