10 Random Evolutions In the previous chapters, we have examined stochastic models whose path functions are either of the jump variety or related to Brownian motion--which is continuous but has infinite velocity. The aim of this chapter is to introduce a class of continuous parameter processes, which move in a piecewise linear fashion and whose slopes jump at the times of a Poisson process. The transition probabilities satisfy a system of linear partial differential equations. In the simplest case, the components of the system satisfy the one-dimensional telegraph equation, which was studied by Mark Kac 1 and Sidney Goldstein 2 in the 1950s. 10.1 Two-State Velocity Model We begin with the simplest case of random evolution, based on a set of two real num- bers v 0 = 1, v 1 = -1, which are interpreted as velocities. Meanwhile, we introduce a probability space ( , F , Pr) on which is defined a sequence of independent random variables with the common exponential distribution -t