Boundary-crossing probabilities for the Brownian motion and poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test

摘要: Let w(t), 0 ≦ t ∞, be a Brownian motion process, i.e., zero-mean separable normal process with Pr{w(0) = 0} 1, E{w(t 1)w(t 2)}= min (t 2), and let a, b denote the boundaries defined by y a(t), b(t), where b(0) < a(0) b(t) T ∞. A basic problem in many fields such as diffusion theory, gambler's ruin, collective risk, Kolmogorov-Smirnov statistics, cumulative-sum methods, sequential analysis optional stopping is that of calculating probability sample path w(t) crosses or before T. This paper shows how this may computed for sufficiently smooth numerical solution integral equations first-passage distribution functions. The technique used to approximate linear recursions whose coefficients are estimated linearising within subintervals. results extended cover tied-down subject condition w(1) 0. Some related Poisson function given. procedures suggested exemplified numerically, first computing particular curved boundary which true known, secondly finite-sample asymptotic powers test against shift mean exponential distribution.