The First Passage Time Problem

Abstract

Thus far we have been concerned with the problem of determining the statistical properties of a stochastic process X(t). We have regarded time as a parameter while X(t), for each fixed t, was a random variable. However, as mentioned in Ch. 1, there are cases in which a certain subset of the state space is assigned and one asks questions concerning the time necessary for the stochastic process to enter this subset for the first time. In a way, this amounts to switching from the description of X(t) to the description of t(X). More specifically, let us consider a single state S (accessible to the process X(t))and ask for the probability distribution of the time T at which X(t) enters state S for the first time. Clearly T is a random variable taking values in the interval (t_o,∞), where t_o is the initial time. Fig. 1.1 shows the times t₁, t₂ and t₃ at which three sample paths of the process X(t), with X(t_o) = x_o , reach S for the first time.