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Approximations of boundary crossing probabilities for a Brownian motion

Part of: Markov processes Numerical approximation and computational geometry Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Alex Novikov ,
Volf Frishling  and
Nino Kordzakhia
Alex Novikov*
Affiliation:
The University of Newcastle, Australia, and Steklov Mathematical Institute, Russia
Volf Frishling*
Affiliation:
Commonwealth Bank of Australia
Nino Kordzakhia*
Affiliation:
The University of Newcastle, Australia
*
∗ Postal address: Department of Statistics, The University of Newcastle, University Drive, Callaghan, NSW 2308, Australia.
∗∗∗ Postal address: 175 Pitt str., Treasury Dealing Department, Commonwealth Bank of Australia, Sydney, NSW 2000, Australia.
∗ Postal address: Department of Statistics, The University of Newcastle, University Drive, Callaghan, NSW 2308, Australia.
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Abstract

Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.

Keywords

First passage timesWerner processGirsanov transformationnumerical integration

MSC classification

Primary: 60J65: Brownian motion
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 65D30: Numerical integration

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 4 , December 1999 , pp. 1019 - 1030
Copyright
Copyright © Applied Probability Trust 1999 

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