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First-passage time of Markov processes to moving barriers

Published online by Cambridge University Press:  14 July 2016

Henry C. Tuckwell  and
Frederic Y. M. Wan
Henry C. Tuckwell*
Affiliation:
Monash University
Frederic Y. M. Wan*
Affiliation:
University of British Columbia
*
∗Postal address: Department of Mathematics, Monash University, Clayton, VIC 3168, Australia.
∗∗Present address: Applied Mathematics Program, FS-20, University of Washington, Seattle, WA 98195, USA.
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Abstract

The first-passage time of a Markov process to a moving barrier is considered as a first-exit time for a vector whose components include the process and the barrier. Thus when the barrier is itself a solution of a differential equation, the theory of first-exit times for multidimensional processes may be used to obtain differential equations for the moments and density of the first-passage time of the process to the barrier. The procedure is first illustrated for first-passage-time problems where the solutions are known. The mean first-passage time of an Ornstein–Uhlenbeck process to an exponentially decaying barrier is then found by numerical solution of a partial differential equation. Extensions of the method to problems involving Markov processes with discontinuous sample paths and to cases where the process is confined between two moving barriers are also discussed.

Keywords

EXIT TIMESDIFFUSION PROCESSNEURAL FIRINGORNSTEIN–UHLENBECK PROCESS
Type
Research Papers
Information
Journal of Applied Probability , Volume 21 , Issue 4 , December 1984 , pp. 695 - 709
Copyright
Copyright © Applied Probability Trust 1984 

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Footnotes

Research partly supported by NSERC of Canada Operating Grant No. A9259 and by U.S. NSF Grant No. MCS-8306592.

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