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

Chuchu Chen, Xinyu Chen, Jialin Hong

TL;DR

This work proves that the law of the iterated logarithm (LIL) for time-homogeneous Markov processes with a unique invariant measure is preserved by numerical approximations using decreasing time steps. The authors overcome non-homogeneity by extracting a quasi-uniform subsequence from the time grid and constructing a predominant martingale along it, with remainder terms vanishing. They establish both $L^2$ and almost-sure limits for the martingale differences and then prove the LIL for the non-homogeneous scheme under verifiable moment and contraction conditions on the process and the step sizes. The results apply to a broad class of stochastic systems, including SODEs and SPDEs, and are demonstrated through concrete SODE and SPDE examples with explicit LIL constants, highlighting substantial practical relevance for long-time numerical analysis of stochastic dynamics.

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.

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

TL;DR

This work proves that the law of the iterated logarithm (LIL) for time-homogeneous Markov processes with a unique invariant measure is preserved by numerical approximations using decreasing time steps. The authors overcome non-homogeneity by extracting a quasi-uniform subsequence from the time grid and constructing a predominant martingale along it, with remainder terms vanishing. They establish both and almost-sure limits for the martingale differences and then prove the LIL for the non-homogeneous scheme under verifiable moment and contraction conditions on the process and the step sizes. The results apply to a broad class of stochastic systems, including SODEs and SPDEs, and are demonstrated through concrete SODE and SPDE examples with explicit LIL constants, highlighting substantial practical relevance for long-time numerical analysis of stochastic dynamics.

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.

Paper Structure

This paper contains 14 sections, 11 theorems, 212 equations.

Table of Contents

  1. Introduction
  2. Related works on numerical limit theorems
  3. Outline
  4. General notation
  5. LIL for the time-homogeneous Markov process
  6. LIL for the non-homogeneous approximation
  7. Construction of the numerical approximation
  8. LIL for the non-homogeneous approximation
  9. Estimates of the martingale difference sequence
  10. Proofs of Propositions in \ref{['sec_numLIL']}
  11. Examples
  12. SODE
  13. SPDE
  14. Proof of Proposition \ref{['thm_exactLIL']}

Key Result

Proposition 2.1

Let a1 hold, $p\in [1,r]$ such that $2p\tilde{r} +4(1+\beta)\gamma_1\leq r$ and $2\tilde{r}(p\tilde{r}+(2+3\beta)\gamma_1) \leq r$. Then $\{X^x_t\}_{t\ge 0}$ admits a unique invariant measure $\mu\in\mathcal{P}(E),$ and fulfills the LIL: For any $f\in\mathcal{C}_{p,\gamma_1},$ it holds that where $v:= \sqrt{2\mu((f-\mu(f))\int_{0}^{\infty}(P_{t}f-\mu(f))\mathrm dt) }$.

Theorems & Definitions (22)

  • Proposition 2.1
  • Theorem 3.1
  • Remark 1
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • proof : Proof of \ref{['thm_numerLIL']}
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • ...and 12 more