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The first-passage density of the Brownian motion process to a curved boundary

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

J. Durbin  and
D. Williams
J. Durbin*
Affiliation:
University of Cambridge
D. Williams*
Affiliation:
University of Cambridge
*
∗Postal address: 31 Southway, London NW11 6RX, UK.
∗∗Postal address: Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, UK.
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Abstract

An expression for the first-passage density of Brownian motion to a curved boundary is expanded as a series of multiple integrals. Bounds are given for the error due to truncation of the series when the boundary is wholly concave or wholly convex. Extensions to the Brownian bridge and to continuous Gauss–Markov processes are given. The series provides a practical method for calculating the probability that a sample path crosses the boundary in a specified time-interval to a high degree of accuracy. A numerical example is given.

Keywords

BROWNIAN BRIDGECONTINUOUS GAUSS–MARKOV PROCESSBOUNDARY-CROSSING

MSC classification

Primary: 60J65: Brownian motion
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 2 , June 1992 , pp. 291 - 304
Copyright
Copyright © Applied Probability Trust 1992 

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Footnotes

This paper was written while the author was visiting the Statistics and Applied Probability Program, University of California, Santa Barbara.

References

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