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First-passage times of two-dimensional Brownian motion

Published online by Cambridge University Press:  11 January 2017

Steven Kou
Affiliation:
National University of Singapore
Haowen Zhong
Affiliation:
Columbia University
Corresponding
E-mail address:

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.

MSC classification

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2017 

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