Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T19:14:56.692Z Has data issue: false hasContentIssue false
  • English
  • Français

Absorption and first-passage times for a compound Poisson process in a general upper boundary

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  14 July 2016

S. F. L. Gallot
S. F. L. Gallot*
Affiliation:
DSIR Physical Sciences
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers the absorption of a non-decreasing compound Poisson process of finite order in a general upper boundary. The problem is relevant in fields such as risk theory, Kolmogorov–Smirnov statistics and sequential analysis. The probability of absorption and first-passage times are given in terms of a generating function which depends on the boundary only and can be computed readily. Absorption is certain or not as the asymptotic slope of the boundary is greater or less than the expected increase of the process in unit time. The case of the linear boundary is considered in detail.

Keywords

ABSORPTION PROBABILITYPASSAGE TIMESPOISSON PROCESS

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60J75: Jump processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 4 , December 1993 , pp. 835 - 850
Copyright
Copyright © Applied Probability Trust 1993 

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] Altiok, T. (1989) (R, r) production/inventory systems. Operat. Res. 37, 266–276.CrossRefGoogle Scholar
[2] Baxter, G. and Donsker, M. D. (1957) On the distribution of the supremum functional for processes with stationary independent increments. Trans. Amer. Math. Soc. 85, 73–87.CrossRefGoogle Scholar
[3] Borovkov, A. A. (1965) On the first passage time for one class of processes with independent increments. Theory Prob. Appl. 10, 331–334.CrossRefGoogle Scholar
[4] Daniels, H. E. (1963) The Poisson process with curved absorbing boundary. Bull. ISI, 34th Session, Ottawa, 994–1008.Google Scholar
[5] Durbin, J. (1971) Boundary crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test. J. Appl. Prob. 8, 431–453.CrossRefGoogle Scholar
[6] Durbin, J. (1985) The first passage density of a continuous Gaussian process to a general boundary. J. Appl. Prob. 22, 99–122.CrossRefGoogle Scholar
[7] Gallot, S. F. L. (1966) Asymptotic absorption probabilities for a Poisson process. J. Appl. Prob. 3, 445–452.CrossRefGoogle Scholar
[8] Gnedenko, B. V. (1962) The Theory of Probability. Chelsea, New York.Google Scholar
[9] Goldman, M. (1971) On the first passage of the integrated Wiener process. Ann. Math. Statist. 42, 2150–2155.CrossRefGoogle Scholar
[10] Harrison, J. M. (1977) The supremum distribution of a Lévy process with no negative jumps. Adv. Appl. Prob. 9, 417–422.CrossRefGoogle Scholar
[11] Leadbetter, M. R. and Rootzen, H. (1988) Extremal theory for stochastic processes. Ann. Prob. 16, 431–478.CrossRefGoogle Scholar
[12] Lefebvre, M. (1989) First passage densities of a two-dimensional process. SIAM J. Appl. Math. 49, 1514–1523.CrossRefGoogle Scholar
[13] Markushevich, A. I. (1965) Theory of Functions of a Complex Variable. Vols I, II and III. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
[14] Pyke, R. (1959) The supremum and infimum of the Poisson process. Ann. Math. Statist. 30, 568–576.CrossRefGoogle Scholar
[15] Szegö, G. (1939) Orthogonal Polynomials. Amer. Math. Soc. Colloquium Publications, Vol XXIII.Google Scholar
[16] Varma, R. S. (1951) On Appell polynomials. Proc. Amer. Math. Soc. 2, 593–596.CrossRefGoogle Scholar
[17] Whittle, P. (1961) Some exact results for one-sided distribution tests of the Kolmogorov-Smirnov type. Ann. Math. Statist. 32, 499–505.CrossRefGoogle Scholar
[18] Zacks, S. (1991) Distributions of stopping times for Poisson processes with linear boundaries. Commun. Statist.-Stoch. Models 7, 233–242.CrossRefGoogle Scholar