2025/11/07 by Chen, Chuchu, Chen, Xinyu, Hong, Jialin
Business, Management and Accounting · Decision Sciences · Economics, Econometrics and Finance · #Advanced Queuing Theory Analysis #FOS: Mathematics #Numerical Analysis (math.NA) #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications
paper · doi:10.48550/arxiv.2511.05217
openalex publication_date 2025/11/07 · openalex created_date 2025/11/11 · openalex updated_date 2026/07/28
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.