The First Passage Problem for a Continuous Markov Process

Abstract

We give in this paper the solution to the first passage problem for a strongly continuous temporally homogeneous Markov process $X(t).$ If $T = T_{ab}(x)$ is a random variable giving the time of first passage of $X(t)$ from the region $a > X(t) > b$ when $a > X(0) = x > b,$ we develop simple methods of getting the distribution of $T$ (at least in terms of a Laplace transform). From the distribution of $T$ the distribution of the maximum of $X(t)$ and the range of $X(t)$ are deduced. These results yield, in an asymptotic form, solutions to certain statistical problems in sequential analysis, nonparametric theory of "goodness of fit," optional stopping, etc. which we treat as an illustration of the theory.