Unbiased Approximations for Stationary Distributions of McKean-Vlasov SDEs

TL;DR

This paper introduces an unbiased estimator for the stationary distribution of McKean-Vlasov SDEs, overcoming discretization bias through novel Monte Carlo techniques, with proofs and numerical demonstrations across various models.

Contribution

It develops the first unbiased estimator for stationary distributions of MVSDEs, with theoretical proofs and practical numerical experiments.

Findings

Unbiased estimator effectively approximates stationary distributions.

Numerical experiments validate estimator's accuracy across models.

The method reduces bias inherent in traditional discretization schemes.

Abstract

We consider the development of unbiased estimators, to approximate the stationary distribution of Mckean-Vlasov stochastic differential equations (MVSDEs). These are an important class of processes, which frequently appear in applications such as mathematical finance, biology and opinion dynamics. Typically the stationary distribution is unknown and indeed one cannot simulate such processes exactly. As a result one commonly requires a time-discretization scheme which results in a discretization bias and a bias from not being able to simulate the associated stationary distribution. To overcome this bias, we present a new unbiased estimator taking motivation from the literature on unbiased Monte Carlo. We prove the unbiasedness of our estimator, under assumptions. In order to prove this we require developing ergodicity results of various discrete time processes, through an appropriate discretization scheme, towards the invariant measure. Numerous numerical experiments are provided, on a range of MVSDEs, to demonstrate the effectiveness of our unbiased estimator. Such examples include the Currie-Weiss model, a 3D neuroscience model and a parameter estimation problem.