Hostname: page-component-7479d7b7d-q6k6v Total loading time: 0 Render date: 2024-07-08T11:45:51.515Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundaries with negative jumps for the Brownian motion

Published online by Cambridge University Press:  14 July 2016

C. Park
C. Park*
Affiliation:
Miami University
*
∗ Postal address: Department of Mathematics and Statistics, Miami University, Bachelor Hall, Oxford, OH 45056, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

First-exit-time problems for Brownian motion have been studied extensively because of their theoretical importance as well as their practical applications. Except for a very few special cases such as straight-line barriers, the distribution (or density) of the first-exit time cannot be expressed in a closed form. In general the distribution and the density appear as solutions of Volterra integral equations. To solve such an equation analytically, some regularity conditions are needed for the barrier function including differentiability. This paper gives various integral equations involving the distribution and the density of the first-exit time for any sectionally continuous barrier, and then shows how to solve those integral equations numerically.

Keywords

FIRST-EXIT TIMEINTEGRAL EQUATIONSSECTIONALLY CONTINUOUS BARRIERWIENER PROCESS
Type
Short Communications
Information
Journal of Applied Probability , Volume 22 , Issue 4 , December 1985 , pp. 964 - 970
Copyright
Copyright © Applied Probability Trust 1985 

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] Breiman, L. (1966) First exit times from a square root boundary. Proc. 5th Berkeley Symp. Math. Statist. Prob. 2(2), 9–16.Google Scholar
[2] Daniels, H. E. (1964) An approximating technique for a curved boundary problem. Adv. Appl. Prob. 6, 194–196.Google Scholar
[3] 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
[4] Ferebee, B. (1982) The tangent approximation to one-sided Brownian exit densities. Z. Wahrscheinlichkeitsth. 61, 309–326.Google Scholar
[5] Ferebee, B. (1983) An asymptotic expansion for one-sided Brownian exit densities. Z. Wahrscheinlichkeitsth. 63, 1–15.Google Scholar
[6] Jennen, C. and Lerche, H. R. (1981) First-exit densities of Brownian motion through one-sided moving boundaries. Z. Wahrscheinlichkeitsth. 55, 133–148.CrossRefGoogle Scholar
[7] Mehr, C. B. and Mcfadden, J. A. (1965) Certain properties of Gaussian processes and their first-passage times. J. R. Statist. Soc. B 27, 505–522.Google Scholar
[8] Park, C. and Beekman, J. A. (1983) Stochastic barriers for the Wiener process. J. Appl. Prob. 20, 338–348.Google Scholar
[9] Park, C. and Paranjape, S. R. (1974) Probabilities of Wiener paths crossing differentiable curves. Pacific J. Math. 53, 579–583.CrossRefGoogle Scholar
[10] Park, C. and Schuurmann, F. J. (1976) Evaluations of barrier-crossing probabilities of Wiener paths. J. Appl. Prob. 13, 267–275.Google Scholar
[11] Park, C. and Schuurmann, F. J. (1983) Partial barrier-absorption probabilities for the Wiener process. J. Appl. Prob. 20, 103–110.Google Scholar
[12] Robbins, H. and Siegmund, D. (1970) Boundary crossing probabilities for the Wiener process and sample sums. Ann. Math. Statist. 41, 1410–1429.Google Scholar
[13] Smith, C. S. (1972) A note on boundary-crossing probabilities for the Brownian motion. J. Appl. Prob. 9, 857–861.CrossRefGoogle Scholar
[14] 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