Extremal behavior of a coupled continuous time random walk

Coupled continuous time random walks (CTRW) model normal and anomalous diffusion of random walkers by taking the sum of random jump lengths dependent on the random waiting times immediately preceding each jump. They are used to simulate diffusion-like processes in econophysics such as stock market fluctuations, where jumps represent financial market microstructure like log-returns. In this and many other applications, the magnitude of the largest observations (e.g. a stock market crash) is of considerable importance in quantifying risk. We use a stochastic process called a coupled continuous time random maxima (CTRM) to determine the density govern∗Corresponding Author Email addresses: [email protected] (Rina Schumer), [email protected] (Boris Baeumer), [email protected] (Mark M. Meerschaert) Partially supported by NSF grants EAR-0120914 and EAR-0817073. Partially supported by NSF grants DMS-0803360 and EAR-0823965. Preprint submitted to Elsevier April 8, 2010 ing the maximum jump length of a particle undergoing a CTRW. CTRM are similar to continuous time random walks but track maxima instead of sums. The many ways in which observations can depend on waiting times can produce an equally large number of CTRM governing density shapes. We compare densities governing coupled CTRM with their uncoupled counterparts for three simple observation/wait dependence structures.