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First-passage times of two-dimensional Brownian motion

Part of: Markov processes

Published online by Cambridge University Press:  11 January 2017

Steven Kou  and
Haowen Zhong
Steven Kou*
Affiliation:
National University of Singapore
Haowen Zhong*
Affiliation:
Columbia University
*
* Postal address: Risk Management Institute and Department of Mathematics, National University of Singapore, 21 Heng Mui Keng Terrace, I3 Building 04-03, Singapore 119613. Email address: [email protected]
** Postal address: Department of Industrial Engineering and Operations Research, Columbia University, 500 West 120th Street, New York, NY 10027, USA.
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Abstract

First-passage times (FPTs) of two-dimensional Brownian motion have many applications in quantitative finance. However, despite various attempts since the 1960s, there are few analytical solutions available. By solving a nonhomogeneous modified Helmholtz equation in an infinite wedge, we find analytical solutions for the Laplace transforms of FPTs; these Laplace transforms can be inverted numerically. The FPT problems lead to a class of bivariate exponential distributions which are absolute continuous but do not have the memoryless property. We also prove that the density of the absolute difference of FPTs tends to ∞ if and only if the correlation between the two Brownian motions is positive.

Keywords

First-passage timestwo-dimensional Brownian motiondefault correlation

MSC classification

Primary: 60J65: Brownian motion
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 4 , December 2016 , pp. 1045 - 1060
Copyright
Copyright © Applied Probability Trust 2017 

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