Unbiased approximation of the ergodic measure for piecewise $\alpha$-stable Ornstein-Uhlenbeck processes arising in queueing networks

TL;DR

This paper introduces an unbiased Euler-Maruyama scheme with decreasing step size for approximating ergodic measures of piecewise alpha-stable Ornstein-Uhlenbeck processes in queueing networks, achieving optimal convergence rates and supporting practical computation.

Contribution

It develops a bias-free numerical approximation method with proven optimal convergence for complex Ornstein-Uhlenbeck processes in queueing models.

Findings

The EM scheme achieves a rate of η^{1/α}_n in Wasserstein-1 distance.

The convergence rate is shown to be optimal via classical OU process analysis.

The paper establishes CLT and MDP for the empirical measures of these processes.

Abstract

Piecewise $\alpha$-stable Ornstein-Uhlenbeck (OU) processes arising in queue networks usually do not have an explicit dissipation, which makes the related numerical methods such as Euler-Maruyama (EM) scheme more difficult to analyze. We develop an EM scheme with decreasing step size $\Lambda=(\eta_n)_{n\in \mathbb{N}}$ to approximate their ergodic measures. This approximation does not have a bias and has a rate $\eta^{1/\alpha}_n$ in Wasserstein-1 distance. We show by the classical OU process that our convergence rate is optimal. We further prove the central limit theorem (CLT) and moderate derivation principle (MDP) for the empirical measure of these piecewise $\alpha$-stable Ornstein-Uhlenbeck processes. In addition, we use the Sinkhorn--Knopp algorithm to compute the Wasserstein-1 distance and conduct simulations for several concrete examples.