Recursive computation of the invariant distributions of Feller processes: Revisited examples and new applications

TL;DR

This paper revisits the recursive methods for computing invariant distributions of Feller processes, applying them to Brownian diffusions with Milstein schemes and censored jump diffusions with Euler schemes, demonstrating convergence under certain conditions.

Contribution

It extends the abstract framework to new applications involving Milstein and Euler schemes for specific diffusion processes, providing convergence proofs.

Findings

Convergence of Milstein scheme for polynomial and exponential test functions.

Convergence of Euler scheme for polynomial growth test functions.

Application of framework to processes with censored jumps.

Abstract

In this paper, we show that the abstract framework developed in Pages & Rey (2017) and inspired by Lamberton & Pages (2002) can be used to build invariant distributions for Brownian diffusion processes using the Milstein scheme and for diffusion processes with censored jump using the Euler scheme. Both studies rely on a weakly mean reverting setting for both cases. For the Milstein scheme we prove the convergence for test functions with polynomial (Wasserstein convergence) and exponential growth. For the Euler scheme of diffusion processes with censored jump we prove the convergence for test functions with polynomial growth.