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December, 1953 The First Passage Problem for a Continuous Markov Process
D. A. Darling, A. J. F. Siegert
Ann. Math. Statist. 24(4): 624-639 (December, 1953). DOI: 10.1214/aoms/1177728918


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.


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D. A. Darling. A. J. F. Siegert. "The First Passage Problem for a Continuous Markov Process." Ann. Math. Statist. 24 (4) 624 - 639, December, 1953.


Published: December, 1953
First available in Project Euclid: 28 April 2007

zbMATH: 0053.27301
MathSciNet: MR58908
Digital Object Identifier: 10.1214/aoms/1177728918

Rights: Copyright © 1953 Institute of Mathematical Statistics

Vol.24 • No. 4 • December, 1953
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