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The joint density of the maximum and its location for a Wiener process with drift

Published online by Cambridge University Press:  14 July 2016

L. A. Shepp
L. A. Shepp*
Affiliation:
Bell Laboratories, Murray Hill, New Jersey
*
∗Postal address: Bell Laboratories, 600 Mountain Ave., Murray Hill, N.J. 07984, U.S.A.
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Abstract

We give a simple expression for the joint probability density of: (a) the maximum value Y = max [X(t), 0 ≦ t ≦ T); (b) its location ; (c) the endpoint X(T), where X(t) = Xc(t) is a Wiener process with drift, Xc(t) = W(t) + ct, 0 ≦ t ≦ T. That is, we find the density p(θ, y, x) = p(θ, y, x; c, T) of , Y, X(T), p(θ, y, x; , Xc(T) ∈ dx) is given by, 0 < θ < T, x ≦ y, 0 < y,

Keywords

WIENER PROCESSLOCATION OF THE MAXIMUMJOINT DENSITY
Type
Short Communications
Information
Journal of Applied Probability , Volume 16 , Issue 2 , June 1979 , pp. 423 - 427
Copyright
Copyright © Applied Probability Trust 1979 

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Footnotes

†

The problem was posed by the economists C. A. Futia, M. B. Goldman, and H. B. Sosin.

References

[1] Cameron, R. H. and Martin, W. T. (1944) Transformations of Wiener integrals under translations. Ann. Math. 45, 386–396.CrossRefGoogle Scholar
[2] Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
[3] Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. 1. 2nd edn. Wiley, New York.Google Scholar
[4] Levy, P. (1965) Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris.Google Scholar
[5] Oberhettinger, F. and Badii, L. (1973) Tables of Laplace Transforms. Springer-Verlag, Berlin.CrossRefGoogle Scholar
[6] Shepp, L. A. (1966) Radon-Nikodym derivatives of Gaussian measures. Ann. Math. Statist. 37, 321–354.CrossRefGoogle Scholar