Approximation of stationary solutions of Gaussian driven Stochastic Differential Equations

TL;DR

This paper investigates the convergence of Euler scheme empirical measures for non-Markovian Gaussian-driven SDEs, establishing their stationary solutions and analyzing dependence properties.

Contribution

It introduces a method to approximate stationary solutions of Gaussian-driven SDEs and explores their dependence structure, extending understanding beyond Markovian cases.

Findings

Empirical measures converge to stationary solutions.

Stationary solutions depend on initial values and Gaussian process.

Under certain conditions, past and future of the process are independent.

Abstract

We study sequences of empirical measures of Euler schemes associated to some non-Markovian SDEs: SDEs driven by Gaussian processes with stationary increments. We obtain the functional convergence of this sequence to a stationary solution to the SDE. Then, we end the paper by some specific properties of this stationary solution. We show that, in contrast to Markovian SDEs, its initial random value and the driving Gaussian process are always dependent. However, under an integral representation assumption, we also obtain that the past of the solution is independent to the future of the underlying innovation process of the Gaussian driving process.