Ergodic approximation of the distribution of a stationary diffusion : rate of convergence

TL;DR

This paper extends CLTs to Lipschitz functionals of ergodic diffusions and their Euler schemes, providing convergence rates and illustrating applications in barrier option pricing.

Contribution

It introduces a broad class of CLTs for Lipschitz functionals of diffusions and their discretizations, with convergence rate analysis.

Findings

CLTs hold for Lipschitz functionals of diffusions and Euler schemes

Convergence rates are established for these CLTs

Simulations demonstrate applications in barrier option pricing

Abstract

We extend to Lipschitz continuous functionals either of the true paths or of the Euler scheme with decreasing step of a wide class of Brownian ergodic diffusions, the Central Limit Theorems formally established for their marginal empirical measure of these processes (which is classical for the diffusions and more recent as concerns their discretization schemes). We illustrate our results by simulations in connection with barrier option pricing.