The First Passage Problem for a Continuous Markov Process

We give in this paper the solution to the first passage problem for a strongly continuous temporally homogeneous Markov process X (t). If T = T₀₁ (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.