Recursive computation of the invariant distribution of a diffusion

Abstract

We propose a recursive stochastic algorithm with decreasing step to compute the invariant distribution $v$ of a Brownian diffusion process, in which we approximate $ν(f)$ for a wide class of possibly unbounded continuous functions $f$. We consider a somewhat general setting which includes cases where the diffusion may have several invariant distributions. Our main convergence result contains as a corollary the almost sure central limit theorem. Further, we investigate the weak rate of convergence of the algorithm. We show, in the class of polynomial steps$\gamma_n=n^{-\alpha}$, that it can be at most $n^{1/3}$ when the white noise has third moment zero and $n^{1/4}$ otherwise, where $n$ denotes the number of iterations of the algorithm.