Triangular array limits for continuous time random walks

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:118

Article

Meerschaert, Mark M.¹  Scheffler, Hans-Peter² 

[1] Michigan State Univ, Dept Stat & Probabil, E Lansing, MI 48824 USA

[2] Univ Siegen, Fachbereich Math, D-57068 Siegen, Germany

关键词: continuous time random walk;    subordinator;    hitting timed;    fractional Cauchy problem;   

DOI  :  10.1016/j.spa.2007.10.005

来源: Elsevier

【 摘 要 】

A continuous time random walk (CTRW) is a random walk subordinated to a renewal process, used in physics to model anomalous diffusion. Transition densities of CTRW scaling limits solve fractional diffusion equations. This paper develops more general limit theorems, based oil triangular arrays, for sequences of CTRW processes. The array elements consist of random vectors that incorporate both the random walk jump variable and the waiting time preceding that jump. The CTRW limit process consists of a vector-valued Levy process whose time parameter is replaced by the hitting time process of a real-valued nondecreasing Levy process (subordinator). We provide a formula for the distribution of the CTRW limit process and show that their densities solve abstract space-time diffusion equations. Applications to finance are discussed, and a density formula for the hitting time of any strictly increasing subordinator is developed. (C) 2007 Elsevier B. V. All rights reserved.

【 授权许可】

Free