Triangular array limits for continuous time random walks

• Published in: Stochastic processes and their applications 2008-09, Vol.118 (9), p.1606-1633
• Main Authors: Meerschaert, Mark M., Scheffler, Hans-Peter
• Format: Article
• Language: English
• Subjects: Continuous time random walk; Continuous time random walk Subordinator Hitting time Fractional Cauchy problem; Exact sciences and technology; Fractional Cauchy problem; Hitting time; Limit theorems; Mathematics; Probability and statistics; Probability theory and stochastic processes; Probability theory on algebraic and topological structures; Sciences and techniques of general use; Stochastic analysis; Stochastic processes; Subordinator
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/j.spa.2007.10.005

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 on 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 Lévy process whose time parameter is replaced by the hitting time process of a real-valued nondecreasing Lévy 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.