Fast $L_2$-approximation of integral-type functionals of Markov processes

TL;DR

This paper develops improved strong $L_2$-approximation rates for integral functionals of Markov processes using transition densities, matching the accuracy previously achieved for one-dimensional diffusions.

Contribution

It introduces an enhanced method for $L_2$-approximation of Markov process functionals based solely on transition densities, extending previous results.

Findings

Achieves optimal $L_2$-approximation rates for Markov processes

Method applies under minimal assumptions on transition densities

Matches accuracy levels known for one-dimensional diffusions

Abstract

In this paper, we provide strong $L_2$-rates of approximation of the integral-type functionals of Markov processes by integral sums. We improve the method developed in [2]. Under assumptions on the process formulated only in terms of its transition probability density, we get the accuracy that coincides with that obtained in [3] for a one-dimensional diffusion process.