Invariant measures of the Milstein method for stochastic differential equations with commutative noise

TL;DR

This paper investigates the Milstein method's ability to approximate invariant measures of stochastic differential equations with commutative noise, demonstrating exponential decay and convergence rates through theoretical analysis and numerical simulations.

Contribution

It provides new theoretical insights into the convergence and decay rates of the Milstein method's invariant measures for stochastic differential equations with commutative noise.

Findings

Exponential decay rate of transition probability kernel

Convergence rate of numerical invariant measure is one

Numerical simulations confirm theoretical results

Abstract

In this paper, the Milstein method is used to approximate invariant measures of stochastic differential equations with commutative noise. The decay rate of the transition probability kernel generated by the Milstein method to the unique invariant measure of the method is observed to be exponential with respect to the time variable. The convergence rate of the numerical invariant measure to the underlying one is shown to be a one. Numerical simulations are presented to demonstrate the theoretical results.