The law of iterated logarithm for numerical approximation of time-homogeneous Markov process

TL;DR

This paper proves that a specific numerical approximation method for time-homogeneous Markov processes preserves the law of the iterated logarithm, ensuring the approximation captures the process's almost sure fluctuation behavior.

Contribution

It establishes the LIL for numerical approximations of Markov processes using a decreasing time-step strategy, addressing an open problem in stochastic numerical analysis.

Findings

LIL holds for the numerical approximation under verifiable assumptions.

The approach applies to SODEs and SPDEs.

A quasi-uniform time-grid subsequence is key to the proof.

Abstract

The law of the iterated logarithm (LIL) for the time-homogeneous Markov process with a unique invariant measure characterizes the almost sure maximum possible fluctuation of time averages around the ergodic limit. Whether a numerical approximation can preserve this asymptotic pathwise behavior remains an open problem. In this work, we give a positive answer to this question and establish the LIL for the numerical approximation of such a process under verifiable assumptions. The Markov process is discretized by a decreasing time-step strategy, which yields the non-homogeneous numerical approximation but facilitates a martingale-based analysis. The key ingredient in proving the LIL for such numerical approximation lies in extracting a quasi-uniform time-grid subsequence from the original non-uniform time grids and establishing the LIL for a predominant martingale along it, while the remainder terms converge to zero. Finally, we illustrate that our results can be flexibly applied to numerical approximations of a broad class of stochastic systems, including SODEs and SPDEs.