The backward Euler-Maruyama method for invariant measures of stochastic differential equations with super-linear coefficients

TL;DR

This paper investigates the backward Euler-Maruyama method for approximating invariant measures of stochastic differential equations with super-linear coefficients, proving existence, uniqueness, and convergence of the numerical invariant measure.

Contribution

It establishes the theoretical foundation for the BEM method's effectiveness in handling super-linear growth in SDE coefficients, including proofs of invariant measure properties.

Findings

Proves existence and uniqueness of the numerical invariant measure.

Shows convergence of the numerical measure to the true invariant measure.

Provides simulations demonstrating theoretical results and applications.

Abstract

The backward Euler-Maruyama (BEM) method is employed to approximate the invariant measure of stochastic differential equations, where both the drift and the diffusion coefficient are allowed to grow super-linearly. The existence and uniqueness of the invariant measure of the numerical solution generated by the BEM method are proved and the convergence of the numerical invariant measure to the underlying one is shown. Simulations are provided to illustrate the theoretical results and demonstrate the application of our results in the area of system control.