GENERAL DIFFUSION PROCESSES AS LIMIT OF TIME-SPACE MARKOV CHAINS

• 成果类型: Article
• 署名作者: Anagnostakis, Alexis; Lejay, Antoine; Villemonais, Denis
• 署名单位: Universite de Lorraine; Inria; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1902
• 发表日期: 2023
• 页码: 3620-3651
• 关键词: sticky; approximation; CONTRACTS

摘要

We prove the convergence of the law of grid-valued random walks, which can be seen as time-space Markov chains, to the law of a general diffusion process. This includes processes with sticky features, reflecting or absorbing boundaries and skew behavior. We prove that the convergence occurs at any rate strictly inferior to (1/4) perpendicular to (1/p) in terms of the maximum cell size of the grid, for any p-Wasserstein distance. We also show that it is possible to achieve any rate strictly inferior to (1/2) perpendicular to (2/p) if the grid is adapted to the speed measure of the diffusion, which is optimal for p <= 4. This result allows us to set up asymptotically optimal approximation schemes for general diffusion processes. Last, we experiment numerically on diffusions that exhibit various features.