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Linear and Nonlinear Boundary Crossing Probabilities for Brownian Motion and Related Processes

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

James C. Fu  and
Tung-Lung Wu
James C. Fu*
Affiliation:
University of Manitoba
Tung-Lung Wu*
Affiliation:
University of Manitoba
*
∗Postal address: Department of Statistics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada.
∗Postal address: Department of Statistics, University of Manitoba, Winnipeg, MB, R3T 2N2, Canada.
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Abstract

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We propose a new method to obtain the boundary crossing probabilities or the first passage time distribution for linear and nonlinear boundaries for Brownian motion. The method also covers certain classes of stochastic processes associated with Brownian motion. The basic idea of the method is based on being able to construct a finite Markov chain, and the boundary crossing probability of Brownian motion is cast as the limiting probability of the finite Markov chain entering a set of absorbing states induced by the boundaries. Error bounds are obtained. Numerical results for various types of boundary studied in the literature are provided in order to illustrate our method.

Keywords

Boundary crossing probabilityfirst passage timeBrownian motiondiffusion processfinite Markov chain imbeddingdiscretizationrandom walk

MSC classification

Secondary: 60J65: Brownian motion 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 60J60: Diffusion processes 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 4 , December 2010 , pp. 1058 - 1071
Copyright
Copyright © Applied Probability Trust 2010 

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