Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-09T01:32:39.628Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic barriers for the Wiener process

Published online by Cambridge University Press:  14 July 2016

C. Park  and
J. A. Beekman
C. Park*
Affiliation:
Miami University
J. A. Beekman*
Affiliation:
Ball State University
*
∗ Postal address: Department of Mathematics and Statistics, Bachelor Hall, Miami University, Oxford, OH 45056, U.S.A.
∗∗ Postal address: Department of Mathematical Sciences, Ball State University, Muncie IN 47306, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The probabilities of the type P[sup0≦t ≦ TW(t) − f(t) ≧ 0] have been extensively studied when f(t) is a deterministic function. This paper discusses the probabilities of the type P{sup0≦t ≦ TW(t) − [f(t) + X(t)] ≧ 0} when X(t) is a stochastic process. By taking compound Poisson processes as X(t), the paper gives procedures for finding such probabilities.

Keywords

STANDARD WIENER PROCESSSTOCHASTIC BARRIERDETERMINISTIC FUNCTIONSTATIONARY INDEPENDENT INCREMENTSSECTIONALLY CONTINUOUSNORMAL DISTRIBUTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 2 , June 1983 , pp. 338 - 348
Copyright
Copyright © Applied Probability Trust 1983 

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] Beekman, J. A. and Fuelling, C. P. (1977) Refined distributions for a multi-risk stochastic process. Scand. Actuarial J., 175–183.CrossRefGoogle Scholar
[2] Beekman, J. A. and Fuelling, C. P. (1979) A multi-risk stochastic process. Trans. Soc. Actuaries 30, 371–397.Google Scholar
[3] Beekman, J. A. and Fuelling, C. P. (1980) Simulation of a multi-risk collective model. In Proc. 1975 Brown Actuarial Research Conference. Academic Press, New York.Google Scholar
[4] Beekman, J. A. and Fuelling, C. P. (1981) Risk convolution calculations. Scand. Actuarial J., 151–164.Google Scholar
[5] Doob, J. L. (1949) Heuristic approach to the Kolmogorov-Smirnov theorem. Ann. Math. Statist. 20, 393–403.Google Scholar
[6] Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
[7] 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.Google Scholar
[8] Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
[9] Gerber, H. U. (1970) An extension of the renewal equation and its application in the collective theory of risk. Skand. Aktuarietidskr. 53, 205–210.Google Scholar
[10] Park, C. and Paranjape, S. R. (1974) Probabilities of Wiener paths crossing differentiable curves. Pacific J. Math. 50, 579–583.Google Scholar
[11] Park, C. and Schuurmann, F. J. (1976) Evaluations of barrier-crossing probabilities of Wiener paths. J. Appl. Prob. 13, 267–275.Google Scholar
[12] Park, C. and Schuurmann, F. J. (1980) Evaluations of absorption probabilities for the Wiener process on large intervals. J. Appl. Prob. 17, 363–372.Google Scholar
[13] Pyke, R. (1959) The supremum and infimum of the Poisson process. Ann. Math. Statist. 30, 568–576.Google Scholar