Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T17:33:07.985Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on boundary - crossing probabilities for the Brownian motion

Published online by Cambridge University Press:  14 July 2016

C. S. Smith
C. S. Smith*
Affiliation:
London School of Economics and Political Science
Get access Rights & Permissions [Opens in a new window]

Abstract

In his paper ‘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), Durbin derives an integral equation whose kernel has a singularity. Since direct solution of an approximating set of simultaneous equations would be very inaccurate, he uses probability arguments to approximate to integrals of sub-intervals. In this note, two alternative procedures are discussed. One makes a linear transformation of the original integral equation to eliminate the singularity; the other, due to Weiss and Anderssen, integrates the singular factor in the kernel over the sub-interval. Computation of a special case indicates this latter method to be the most effective of the three.

Keywords

BOUNDARY-CROSSING PROBABILITIESGENERALIZED ABEL EQUATIONNUMERICAL SOLUTIONINTEGRAL EQUATIONSINGULAR KERNEL
Type
Short Communications
Information
Journal of Applied Probability , Volume 9 , Issue 4 , December 1972 , pp. 857 - 861
Copyright
Copyright © Applied Probability Trust 1972 

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

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
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
Hildebrand, E. B. (1956) Introduction to Numerical Analysis. McGraw-Hill, New York.Google Scholar
Kowalewski, G. (1930) Integralgleichungen, Walter de Gruyter & Co., Berlin.Google Scholar
Weiss, R. (1971) Product integration for the generalized Abel equation. Math. Comp. To appear.CrossRefGoogle Scholar
Weiss, R. and Anderssen, R. S. (1971) A product integration method for a class of singular first-kind Volterra equations. Numer. Math. To appear.CrossRefGoogle Scholar