Approximation of stationary solutions to SDEs driven by multiplicative fractional noise

TL;DR

This paper investigates how to approximate stationary solutions of stochastic differential equations driven by multiplicative fractional Brownian noise, establishing convergence properties of Euler scheme-based empirical measures.

Contribution

It extends previous work on additive noise to the case of multiplicative fractional noise, providing convergence results for Euler scheme approximations.

Findings

Established convergence of empirical measures to stationary solutions.

Extended ergodic analysis to SDEs with fractional multiplicative noise.

Provided theoretical foundations for numerical approximation of such SDEs.

Abstract

In a previous paper, we studied the ergodic properties of an Euler scheme of a stochastic differential equation with a Gaussian additive noise in order to approximate the stationary regime of such equation. We now consider the case of multiplicative noise when the Gaussian process is a fractional Brownian Motion with Hurst parameter H>1/2 and obtain some (functional) convergences properties of some empirical measures of the Euler scheme to the stationary solutions of such SDEs.