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Crossing problems for non-constant thresholds and certain non-Markov processes

Published online by Cambridge University Press:  14 July 2016

Barry Belkin ,
Lawrence D. Stone  and
Martin Avery Snyder
Barry Belkin
Affiliation:
Daniel H. Wagner, Associates, Paoli, Pennsylvania
Lawrence D. Stone
Affiliation:
Daniel H. Wagner, Associates, Paoli, Pennsylvania
Martin Avery Snyder
Affiliation:
Bryn Mawr College, Pennsylvania
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Abstract

A result in Stone, Belkin, and Snyder ((1970) J. Math. Anal. Appl.30, 448–470) gave a method for finding the Laplace-Stieltjes transform of the distribution of certain non-negative, homogeneous, additive functionals of a Markov process with stationary transition measure. By considering certain two dimensional Markov processes and applying this result, a method is obtained for finding time above a threshold and first passage distributions for a one dimensional process either when (1) the process is Markovian and the threshold is possibly non-constant, or (2) the threshold is constant and the process is the indefinite integral of a Markov process. Specific process-threshold combinations are considered in several examples including the case of a linear threshold for a Wiener process and a for compound Poisson process with exponential (either one-sided or bilateral) after-jump distribution. In addition, the first passage distribution to a constant threshold is computed for an integrated Poisson sampling process.

Keywords

FIRST PASSAGE PROBLEMSMARKOV PROCESSESNON-CONSTANT THRESHOLD CROSSINGDISTRIBUTIONS OF FUNCTIONALS OF MARKOV PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 4 , December 1972 , pp. 769 - 788
Copyright
Copyright © Applied Probability Trust 1972 

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