The Annals of Mathematical Statistics

The First Passage Problem for a Continuous Markov Process

D. A. Darling and A. J. F. Siegert

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Abstract

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.

Article information

Source
Ann. Math. Statist., Volume 24, Number 4 (1953), 624-639.

Dates
First available in Project Euclid: 28 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177728918

Digital Object Identifier
doi:10.1214/aoms/1177728918

Mathematical Reviews number (MathSciNet)
MR58908

Zentralblatt MATH identifier
0053.27301

JSTOR
links.jstor.org

Citation

Darling, D. A.; Siegert, A. J. F. The First Passage Problem for a Continuous Markov Process. Ann. Math. Statist. 24 (1953), no. 4, 624--639. doi:10.1214/aoms/1177728918. https://projecteuclid.org/euclid.aoms/1177728918


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