Asymptotically unbiased approximation of the QSD of diffusion processes with a decreasing time step Euler scheme

TL;DR

This paper introduces a recursive algorithm using a decreasing step Euler scheme to accurately approximate the quasistationary distribution of elliptic diffusions, with proven convergence and error bounds.

Contribution

It presents a novel recursive algorithm for QSD approximation with decreasing steps, including convergence proofs and error analysis for diffusion processes.

Findings

Almost sure convergence of the algorithm

Recovery of the survival rate approximation

New bounds on weak error for diffusion with renewal

Abstract

We build and study a recursive algorithm based on the occupation measure of an Euler scheme with decreasing step for the numerical approximation of the quasistationary distribution (QSD) of an elliptic diffusion in a bounded domain. We prove the almost sure convergence of the procedure for a family of redistributions and show that we can also recover the approximation of the rate of survival and the convergence in distribution of the algorithm. This last point follows from some new bounds on the weak error related to diffusion dynamics with renewal.