Inhomogeneous functionals and approximations of invariant distributions of ergodic diffusions: Error analysis through central limit theorem and moderate deviation asymptotics

TL;DR

This paper analyzes the error of Euler discretization schemes for ergodic diffusions, establishing convergence, central limit theorem, and moderate deviation principles to understand their efficiency in estimating invariant distributions.

Contribution

It provides the first detailed error analysis of discretization schemes for invariant distribution estimation, including CLT and moderate deviations, extending understanding beyond trajectory approximation.

Findings

Convergence of the numerical scheme under suitable step size.

Establishment of central limit theorem for the error.

Derivation of moderate deviation principle for the error.

Abstract

The paper considers an Euler discretization based numerical scheme for approximating functionals of invariant distribution of an ergodic diffusion. Convergence of the numerical scheme is shown for suitably chosen discretization step, and a thorough error analysis is conducted by proving central limit theorem and moderate deviation principle for the error term. The paper is a first step in understanding efficiency of discretization based numerical schemes for estimating invariant distributions, which is comparatively much less studied than the schemes used for generating approximate trajectories of diffusions over finite time intervals. The potential applications of these results also extend to other areas including mathematical physics, parameter inference of ergodic diffusions and analysis of multiscale dynamical systems with averaging.