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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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References

[1] Abramowitz, M. and Stegun, I. A. (1964) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U. S. Government Printing Office, Washington, D. C.Google Scholar
[2] Arfwedson, G. (1950) Some problems in the collective theory of risk. Skand. Aktuarietidskr. 33, 1–38.Google Scholar
[3] Belkin, B. (1971) First passage to a general threshold for a process corresponding to sampling at Poisson times. J. Appl. Prob. 8, 573–588.CrossRefGoogle Scholar
[4] Campbell, G. A. and Foster, R. M. (1948) Fourier Integrals. D. Van Nostrand Company, New Jersey.Google Scholar
[5] Dynkin, E. B. (1965) Markov Processes, I. Academic Press, New York.Google Scholar
[6] Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. II. John Wiley, New York.Google Scholar
[7] Mehr, C. B. and Mcfadden, J. A. (1965) Certain properties of Gaussian processes and their first-passage times. J. Roy. Statist. Soc. 27, 505–522.Google Scholar
[8] Takács, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. John Wiley, New York.Google Scholar
[9] Stone, L. A., Belkin, B. and Snyder, M. A. (1970) Distribution of time above a threshold for Markov processes. J. Math. Anal. Appl. 30, 448–470.CrossRefGoogle Scholar
[10] Widder, D. V. (1941) The Laplace Transform. Mathematical Series, Vol. 6. Princeton University Press.Google Scholar