THE FIRST PASSAGE PROBLEM FOR A CONTINUOUS MARKOFF PROCESS,

摘要

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. (Author)