Approximation of invariant probability measures for super-linear stochastic functional differential equations with infinite delay

TL;DR

This paper introduces an explicit truncated Euler-Maruyama scheme for approximating invariant probability measures of super-linear stochastic functional differential equations with infinite delay, ensuring convergence and computational efficiency.

Contribution

It proposes a novel explicit numerical scheme with proven convergence and rate for invariant measures of super-linear SFDEs with infinite delay, overcoming previous implicit and finite-delay limitations.

Findings

The TEM scheme converges strongly over finite horizons.

The numerical IPM is unique and converges to the true IPM.

Explicit convergence rates are established in Wasserstein distance.

Abstract

This paper studies explicit numerical approximations of the invariant probability measures (IPMs) for stochastic functional differential equations (SFDEs) with infinite delay under one-sided Lipschitz condition on the drift coefficient. To date, numerical approximations of IPMs for super-linear SFDEs have been focused to finite-delay cases and implicit schemes that require additional computational effort. To overcome these constraints, we propose an explicit truncated Euler-Maruyama (TEM) scheme employing both time and space truncation for SFDEs with infinite delay, which is explicit and requires only finite historical storage. Firstly, we establish the strong convergence of the numerical segment process and determine its convergence rate over any finite time horizon. Next, we show that the numerical segment process generated by the TEM scheme admits a unique numerical IPM. Leveraging these results, we then prove that the numerical IPM converges to the exact IPM in the Wasserstein distance, with an explicitly obtained convergence rate.