Non-Independent Continuous Time Random Walks

Abstract

The usual development of the continuous time random walk (CTRW) assumes that jumps and time intervals are independent (and identically distributed) random variables. In this paper we address the theoretical setting of non-independent CTRW's where jumps and/or time intervals are correlated. An exact solution to the problem is obtained for the special but relevant case in which the correlation solely depends on the signs of consecutive jumps. Even in this simple case some interesting features arise such as transitions from unimodal to bimodal distributions due to correlation. We also develop the necessary analytical techniques and approximations to handle more general situations that can appear in practice.

Keywords: Stochastic processes, Markov processes, Random walks, CTRW, Approximations