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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
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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 

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References

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