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The Wiener process between a reflecting and an absorbing barrier

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Wolfgang Schwarz
Wolfgang Schwarz*
Affiliation:
Freie Universität Berlin
*
∗Postal address: Freie Universität Berlin, Habelschwerdter Allee 45, D(W)-1000 Berlin 33, Germany.
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Abstract

We consider a Wiener process between a reflecting and an absorbing barrier and derive a series solution for the transition density of the process and for the density of the time to absorption. If the drift is towards the reflecting barrier, the variance is not too large, and the distance of the barriers is not too small, the leading term of the series derives from imaginary solutions of the basic eigenvalue equation of this problem. It is shown that these leading terms often make the dominant contribution to the complete series. Finally, we consider previous attempts by Fürth [3], Cox and Miller [1], and by Goel and Richter-Dyn [5] to solve the stated problem and point out, in some detail, why their solutions are wrong.

Keywords

DIFFUSIONFIRST-PASSSAGE TIMESPECTRAL EXPANSION

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J60: Diffusion processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 3 , September 1992 , pp. 597 - 604
Copyright
Copyright © Applied Probability Trust 1992 

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References

[1] Cox, D. R. and Miller, H. D. (1965) The Theory of Stochastic Processes. Chapman and Hall, London.Google Scholar
[2] Darling, D. A. and Siegert, A. J. F. (1953) The first passage problem for a continuous Markov process. Ann. Math. Statist. 24, 624–639.CrossRefGoogle Scholar
[3] Fürth, R. (1917) Einige Untersuchungen über Brownsche Bewegung an einem Einzelteilchen. Ann. Phys. 53, 177–213.Google Scholar
[4] Gardiner, C. W. (1985) Handbook of Stochastic Methods. Springer-Verlag, Berlin.Google Scholar
[5] Goel, N. S. and Richter-Dyn, N. (1974) Stochastic Models in Biology. Academic Press, New York.Google Scholar
[6] Riemann, B. and Weber, H. (1912) Die partiellen Differentialgleichungen der Mathematischen Physik. Band II. Vieweg, Braunschweig.Google Scholar
[7] Risken, H. (1989) The Fokker-Planck Equation, 2nd ed. Springer-Verlag, Berlin.Google Scholar
[8] Smoluchowski, M. Von (1915) über Brownsche Molekularbewegung unter Einwirkung äußerer Kräfte und deren Zusammenhang mit der verallgemeinerten Diffusionsgleichung. Ann. Phy. 48, 1103–1112.Google Scholar
[9] Sommerfeld, A. (1947) Partielle Differentialgleichungen der Physik. Akademische Verlagsgesellschaft, Leipzig.Google Scholar
[10] Street, R. L. (1973) The Analysis and Solution of Partial Differential Equations. Brooks/Cole, Monterey, CA.Google Scholar
[11] Titchmarsh, E. C. (1962) Eigenfunction Expansions Associated with Second-Order Differential Equations. Clarendon Press, Oxford.Google Scholar
[12] Trim, D. W. (1990) Applied Partial Differential Equations. PWS-Kent, Boston, MA.Google Scholar
[13] Wenocur, M. L. (1988) Diffusion first passage times: approximations and related differential equations. Stoch. Proc. Appl. 27, 159–177.Google Scholar