Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-23T12:19:43.910Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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