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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
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.

References

Handelsman, R. and Keller, J, (1966) Quantum-mechanical second virial coefficient of a hard sphere gas at high temperature. Phys. Rev. 148, 94–97.CrossRefGoogle Scholar
Hogan, ?. (1984) Problems in Boundary Crossings for Random Walks. Ph.D. Thesis, Stanford University.Google Scholar
Jennen, C. and Lerche, R. (1981) First exit densities of Brownian motion through one-sided moving boundaries. Z. Wahrscheinlichkeitsth. 55, 133–148.Google Scholar
Jennen, C. and Lerche, H. R. (1982) Asymptotic densities of stopping times associated with tests of power one. Z. Wahrscheinlichkeitsth. 61, 501–511.Google Scholar
Kac, M. (1966) Can one hear the shape of a drum? Amer. Math. Montly 73, 1–23.CrossRefGoogle Scholar
Kiefer, J. (1959) K-sample analogues of the Kolmogorov-Smirnov and Cramér-von Mises tests. Ann. Math. Statist. 30, 420–447.Google Scholar
Loève, M. (1963) Probability Theory, 3rd edn. Van Nostrand, Princeton.Google Scholar
Louchard, G. (1968) Mouvement brownien et valeurs propres du Laplacien. Ann. Inst. H. Poincaré B 4, 331–342.Google Scholar
Mckean, H. P. Jr and Singer, I. M. (1967) Curvature and the eigenvalues of the Laplacian. J. Diff. Geometry 1, 43–69.Google Scholar
Millman, R. S. and Parker, G. D. (1977) Elements of Differential Geometry. Prentice-Hall, Englewood Cliffs.Google Scholar
Siegmund, D. (1968) On the asymptotic normality of one-sided stopping rules. Ann. Math. Statist. 39, 1493–1497.CrossRefGoogle Scholar
Siegmund, D. (1982) Large deviations for boundary crossing probabilities. Ann. Prob. 10, 581–588.Google Scholar
Siegmund, D. (1985) Sequential Analysis: Tests and Confidence Intervals. Springer-Verlag, New York.CrossRefGoogle Scholar
Smith, L. (1981) The asymptotics of the heat equation for a boundry value problem. Invent. Math. 63, 467–493.Google Scholar
Stewartson, K. and Waechter, R. T. (1971) On hearing the shape of a drum. Proc. Camb. Phil. Soc. 69, 353–363.CrossRefGoogle Scholar