International Journal of Stochastic Analysis
Volume 2011 (2011), Article ID 296259, 5 pages
doi:10.1155/2011/296259
Research Article

Maximizing the Mean Exit Time of a Brownian Motion from an Interval

Mario Lefebvre

Département de Mathématiques et de Génie Industriel, École Polytechnique, C.P. 6079, Succursale Centre-ville, Montréal, QC, H3C 3A7, Canada

Received 31 December 2010; Accepted 20 January 2011

Academic Editor: Peter Kloeden

Copyright © 2011 Mario Lefebvre. 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

Let 𝑋 ( 𝑡 ) be a controlled one-dimensional standard Brownian motion starting from 𝑥 ∈ ( − 𝑑 , 𝑑 ) . The problem of optimally controlling 𝑋 ( 𝑡 ) until | 𝑋 ( 𝑡 ) | = 𝑑 for the first time is solved explicitly in a particular case. The maximal value that the instantaneous reward given for survival in ( − 𝑑 , 𝑑 ) can take is determined.