Approximation of Invariant Measures for Stochastic Differential Equations with Piecewise Continuous Arguments via Backward Euler Method

TL;DR

This paper studies stochastic differential equations with piecewise continuous arguments driven by multiplicative noise, proving the backward Euler method can accurately approximate their invariant measures with convergence order 1.

Contribution

It establishes that the backward Euler method preserves the invariant measure structure of SDEs with PCAs and provides a convergence rate for the numerical invariant measure.

Findings

Numerical solution at integer times is Markovian and admits a unique invariant measure.

The numerical invariant measure converges to the true invariant measure with order 1.

Numerical experiments confirm the theoretical convergence results.

Abstract

For the stochastic differential equation (SDE) which has piecewise continuous arguments (PCAs), is driven by multiplicative noises and its drift coefficients are dissipative, we show that the solution at integer time is a Markov chain and admits a unique invariant measure. In order to inherit numerically the invariant measure of SDE with PCAs, we apply the backward Euler (BE) method to the equation, and prove that the numerical solution at integer time is not only Markovian but also reproduces a unique numerical invariant measure. We present the time-independent weak error analysis for the method under certain hypothesis. Further, we show that the numerical invariant measure converges to the original one with order 1. Numerical experiments verify the theoretical analysis.