Hostname: page-component-76fb5796d-vfjqv Total loading time: 0 Render date: 2024-04-28T20:09:27.172Z Has data issue: false hasContentIssue false
  • English
  • Français

Evaluations of barrier-crossing probabilities of Wiener paths

Published online by Cambridge University Press:  14 July 2016

C. Park  and
F. J. Schuurmann
C. Park
Affiliation:
Miami University, Oxford, Ohio
F. J. Schuurmann
Affiliation:
Miami University, Oxford, Ohio
Get access Rights & Permissions [Opens in a new window]

Abstract

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The main purpose of this paper is to present ways of obtaining probabilities of Wiener paths crossing certain curves on various intervals. The results are extended to other kinds of Gaussian processes.

Keywords

STANDARD WIENER PROCESSCOVARIANCEGAUSSIAN PROCESSESBROWNIAN BRIDGEFIRST HITTING TIME
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 2 , June 1976 , pp. 267 - 275
Copyright
Copyright © Applied Probability Trust 1976 

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] Daniels, H. E. (1969) The minimum of a stationary Markov process superimposed on a U-shaped trend. J. Appl. Prob. 6, 399–408.CrossRefGoogle Scholar
[2] Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
[3] Doob, J. L. (1949) Heuristic approach to the Kolmogorov–Smirnov theorems. Ann. Math. Statist. 20, 393–403.CrossRefGoogle Scholar
[4] 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
[5] Park, C. and Paranjape, S. R. (1974) Probabilities of Wiener paths crossing differentiable curves. Pacific J. Math. 53, 579–583.Google Scholar
[6] Smith, C. S. (1972) A note on boundary-crossing probabilities for the Brownian motion. J. Appl. Prob. 9, 857–861.Google Scholar
[7] Weiss, R. and Anderson, R. S. (1971) A product integration method for a class of singular first-kind Volterra equations. Numer. Math. 18, 442–456.Google Scholar