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Journal of Probability and Statistics
Volume 2015, Article ID 391681, 22 pages
http://dx.doi.org/10.1155/2015/391681
Research Article

On the Computation of the Survival Probability of Brownian Motion with Drift in a Closed Time Interval When the Absorbing Boundary Is a Step Function

Université de Cergy-Pontoise, Laboratoire Thema, 33 boulevard du port, 95011 Cergy-Pontoise Cedex, France

Received 7 April 2015; Accepted 13 May 2015

Academic Editor: Dejian Lai

Copyright © 2015 Tristan Guillaume. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper provides explicit formulae for the probability that an arithmetic or a geometric Brownian motion will not cross an absorbing boundary defined as a step function during a finite time interval. Various combinations of downward and upward steps are handled. Numerical computation of the survival probability is done quasi-instantaneously and with utmost precision. The sensitivity of the survival probability to the number and the ordering of the steps in the boundary is analyzed.