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On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary

Published online by Cambridge University Press:  01 July 2016

Paavo Salminen
Paavo Salminen*
Affiliation:
Åbo Akademi
*
∗Postal address: Åbo Akademi, Mathematical Institute, SF-20500 Åbo 50, Finland.
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Abstract

Let t → h(t) be a smooth function on ℝ+, and B = {Bs; s ≥ 0} a standard Brownian motion. In this paper we derive expressions for the distributions of the variables Th: = inf {S; Bs = h(s)} and λth: = sup {s ≦ t; Bs = h(s)}, where t> 0 is given. Our formulas contain an expected value of a Brownian functional. It is seen that this can be computed, principally, using Feynman–Kac&s formula. Further, we discuss in our framework the familiar examples with linear and square root boundaries. Moreover our approach provides in some extent explicit solutions for the second-order boundaries.

Keywords

AIRY&S FUNCTIONORNSTEIN–UHLENBECK PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 2 , June 1988 , pp. 411 - 426
Copyright
Copyright © Applied Probability Trust 1988 

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