Approximation of Markov semigroups in total variation distance

Abstract

In this paper, we consider Markov chains of the form $X^{n}_{(k+1)/n}=\psi _{k}(X^{n}_{k/n},Z_{k+1}/\sqrt{n},1/n)$ where the innovation comes from the sequence $Z_{k},k\in \mathbb{N} ^{\ast }$ of independent centered random variables with arbitrary law. Then, we study the convergence $\mathbb{E} [f(X^{n}_t)]\rightarrow \mathbb{E} [f(X_t)]$ where $(X_t)_{t \geqslant 0}$ is a Markov process in continuous time. This may be considered as an invariance principle, which generalizes the classical Central Limit Theorem to Markov chains. Alternatively (and this is the main motivation of our paper), $X^{n}$ may be an approximation scheme used in order to compute $\mathbb{E} [f(X_t)]$ by Monte Carlo methods. Estimates of the error are given for smooth test functions $f$ as well as for measurable and bounded $f.$ In order to prove convergence for measurable test functions we assume that $Z_{k}$ satisfies Doeblin’s condition and we use Malliavin calculus type integration by parts formulas based on the smooth part of the law of $Z_{k}$. As an application, we will give estimates of the error in total variation distance for the Ninomiya Victoir scheme.