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On Durbin's Series for the Density of First Passage Times

Part of: Partial differential equations, boundary value problems Partial differential equations on manifolds; differential operators Markov processes

Published online by Cambridge University Press:  14 July 2016

P. Zipkin
P. Zipkin*
Affiliation:
Duke University
*
∗ Postal address: Fuqua School of Business, Duke University, Durham, NC, USA. Email address: paul.zipkin@duke.edu
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Abstract

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Durbin (1992) derived a convergent series for the density of the first passage time of a Weiner process to a curved boundary. We show that the successive partial sums of this series can be expressed as the iterates of the standard substitution method for solving an integral equation. The calculation is thus simpler than it first appears. We also show that, under a certain condition, the series converges uniformly. This strengthens Durbin's result of pointwise convergence. Finally, we present a modified procedure, based on scaling, which sometimes works better. These approaches cover some cases that Durbin did not.

Keywords

First passage timeWiener process

MSC classification

Secondary: 60J65: Brownian motion 65N21: Inverse problems 58J35: Heat and other parabolic equation methods
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 3 , September 2011 , pp. 713 - 722
Copyright
Copyright © Applied Probability Trust 2011 

References

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