PATHWISE OPTIMAL TRANSPORT BOUNDS BETWEEN A ONE-DIMENSIONAL DIFFUSION AND ITS EULER SCHEME

• 成果类型: Article
• 署名作者: Alfonsi, A.; Jourdain, B.; Kohatsu-Higa, A.
• 署名单位: Institut Polytechnique de Paris; Ecole Nationale des Ponts et Chaussees; Ritsumeikan University; Japan Science & Technology Agency (JST)
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/13-AAP941
• 发表日期: 2014
• 页码: 1049-1080
• 关键词: Approximation; distributions; CONVERGENCE; error

摘要

In the present paper, we prove that the Wasserstein distance on the space of continuous sample-paths equipped with the supremum norm between the laws of a uniformly elliptic one-dimensional diffusion process and its Eu ler discretization with N steps is smaller than O(N-2/3+epsilon) where epsilon is an arbitrary positive constant. This rate is intermediate between the strong error estimation in O(N-1/2) obtained when coupling the stochastic differential equation and the Euler scheme with the same Brownian motion and the weak error estimation O(N-1) obtained when comparing the expectations of the same function of the diffusion and of the Euler scheme at the terminal time T. We also check that the supremum over t is an element of [0, T] of the Wasserstein distance on the space of probability measures on the real line between the laws of the diffusion at time t and the Euler scheme at time t behaves like O(root log(N)N-1).