Functional quantization rate and mean regularity of processes with an application to Levy processes

• 成果类型: Article
• 署名作者: Luschgy, Harald; Pages, Gilles
• 署名单位: Universitat Trier; Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Sorbonne Universite
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/07-AAP459
• 发表日期: 2008
• 页码: 427-469
• 关键词: small ball probabilities

摘要

We investigate the connections between the mean pathwise regularity of stochastic processes and their L-r(P)-functional quantization rates as random variables taking values in some L-p ([0, T], dt)-spaces (0 < p <= r). Our main tool is the Haar basis. We then emphasize that the derived functional quantization rate may be optimal (e.g., for Brownian motion or symmetric stable processes) so that the rate is optimal as a universal upper bound. As a first application, we establish the 0((Iog N)(-1/2)) upper bound for general It<(O)over cap> processes which include multidimensional diffusions. Then, we focus on the specific family of Levy processes for which we derive a general quantization rate based on the regular variation properties of its Levy measure at 0. The case of compound Poisson processes, which appear as degenerate in the former approach, is studied specifically: we observe some rates which are between the finite-dimensional and infinite-dimensional usual rates.