THE FIRST PASSAGE PROBLEM FOR A CONTINUOUS MARKOFF PROCESS
Abstract : The solution to the first passage problem for a strongly continuous temporally homogeneous Markoff process X(t) is given. If T = T sub 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, simple methods of getting the distribution of T (at least in terms of a Laplace transform) are developed. 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, non-parametric theory of 'goodness of fit,' optional stopping, etc. which are treated as an illustration of the theory.