Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-24T02:48:53.509Z Has data issue: false hasContentIssue false
  • English
  • Français

Evaluations of absorption probabilities for the Wiener process on large intervals

Published online by Cambridge University Press:  14 July 2016

C. Park  and
F. J. Schuurmann
C. Park*
Affiliation:
Miami University
F. J. Schuurmann*
Affiliation:
Miami University
*
∗Postal address: Department of Mathematics and Statistics, Culler Hall, Miami University, Oxford, OH 45056, U.S.A.
∗Postal address: Department of Mathematics and Statistics, Culler Hall, Miami University, Oxford, OH 45056, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let {W(t), 0≦t<∞} be the standard Wiener process. The computation schemes developed in the past are not computationally efficient for the absorption probabilities of the type P{sup0≦t≦TW(t) − f(t) ≧ 0} when either T is large or f(0) > 0 is small. This paper gives an efficient and accurate algorithm to compute such probabilities, and some applications to other Gaussian stochastic processes are discussed.

Keywords

ABSORPTION PROBABILITYWIENER PROCESSGAUSSIAN STOCHASTIC PROCESSORNSTEIN–UHLENBECK PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 17 , Issue 2 , June 1980 , pp. 363 - 372
Copyright
Copyright © Applied Probability Trust 1980 

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. Actuar. J., 175–183.Google Scholar
[2] Beekman, J. A. and Fuelling, C. P. (1979) A multi-risk stochastic process. Trans. Soc. Actuaries XXX, 371–397.Google Scholar
[3] Doob, J. L. (1949) Heuristic approach to the Kolmogorov–Smirnov theorem. Ann. Math. Statist. 20, 393–403.CrossRefGoogle Scholar
[4] Feller, W. (1957) An Introduction to Probability Theory and its Applications, Vol. 1. Wiley, New York.Google Scholar
[5] Park, C. and Paranjape, S. R. (1974) Probabilities of Wiener paths crossing differentiable curves. Pacific J. Math. 50, 579–583.CrossRefGoogle Scholar
[6] Park, C. and Schuurmann, F. J. (1976) Evaluations of barrier-crossing probabilities of Wiener paths. J. Appl. Prob. 13, 267–275.Google Scholar
[7] Rosenblatt, M. (1962) Random Processes. Oxford University Press, New York.Google Scholar
[8] Shepp, L. A. (1966) Radon–Nikodym derivatives of Gaussian measures. Ann. Math. Statist. 37, 321–354.Google Scholar
[9] Takacs, L. (1967) Combinatorial Methods in the Theory of Stochastic Processes. Wiley, New York.Google Scholar