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Boundaries with negative jumps for the Brownian motion

Published online by Cambridge University Press:  14 July 2016

C. Park
C. Park*
Affiliation:
Miami University
*
∗ Postal address: Department of Mathematics and Statistics, Miami University, Bachelor Hall, Oxford, OH 45056, USA.
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Abstract

First-exit-time problems for Brownian motion have been studied extensively because of their theoretical importance as well as their practical applications. Except for a very few special cases such as straight-line barriers, the distribution (or density) of the first-exit time cannot be expressed in a closed form. In general the distribution and the density appear as solutions of Volterra integral equations. To solve such an equation analytically, some regularity conditions are needed for the barrier function including differentiability. This paper gives various integral equations involving the distribution and the density of the first-exit time for any sectionally continuous barrier, and then shows how to solve those integral equations numerically.

Keywords

FIRST-EXIT TIMEINTEGRAL EQUATIONSSECTIONALLY CONTINUOUS BARRIERWIENER PROCESS
Type
Short Communications
Information
Journal of Applied Probability , Volume 22 , Issue 4 , December 1985 , pp. 964 - 970
Copyright
Copyright © Applied Probability Trust 1985 

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