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Approximate exit probabilities for a Brownian bridge on a short time interval, and applications

Published online by Cambridge University Press:  01 July 2016

H. R. Lerche  and
D. Siegmund
H. R. Lerche*
Affiliation:
University of Freiburg
D. Siegmund*
Affiliation:
Stanford University
*
∗ Postal address: Institut für Mathematische Stochastik, Universität Freiburg i.Br., Hebelstrasse 27, D7800 Freiburg, W. Germany.
∗∗ Postal address: Department of Statistics, Stanford University, Sequoia Hall, Stanford, CA 94305, USA.
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Abstract

Let T be the first exit time of Brownian motion W(t) from a region ℛ in d-dimensional Euclidean space having a smooth boundary. Given points ξ0 and ξ1 in ℛ, ordinary and large-deviation approximations are given for Pr{T < ε |W(0) = ξ0, W(ε) = ξ 1} as ε → 0. Applications are given to hearing the shape of a drum and approximating the second virial coefficient.

Keywords

FIRST PASSAGEHEARING THE SHAPE OF A DRUM

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 1 , March 1989 , pp. 1 - 19
Copyright
Copyright © Applied Probability Trust 1989 

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Footnotes

Research supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 123 at the University of Heidelberg.

Research supported by the Humboldt-Stiftung, Office of Naval Research, and National Science Foundation.

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