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Some remarks on Brownian motion with drift

Published online by Cambridge University Press:  14 July 2016

R. A. Doney and
D. R. Grey
R. A. Doney*
Affiliation:
University of Manchester
D. R. Grey*
Affiliation:
University of Sheffield
*
∗ Postal address: Statistical Laboratory, Department of Mathematics, The University, Manchester M13 0PL, UK.
∗∗ Postal address: Department of Probability and Statistics, The University, Sheffield S3 7RH, UK.
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Abstract

Certain limit theorems due to Berman involve the total time spent by Brownian motion with positive drift below an independent exponentially distributed level. Imhof has calculated the density function and shown that this random variable has two interesting probabilistic properties. We give sample path arguments which explain these two facts.

Keywords

PATH DECOMPOSITIONTIME BELOW A LEVEL

Information

Type
Short Communications
Information
Journal of Applied Probability , Volume 26 , Issue 3 , September 1989, pp. 659 - 663
Copyright
Copyright © Applied Probability Trust 1989 

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References

[1] Berman, S. M. (1982) Sojourns and extremes of a diffusion process on a fixed interval. Adv. Appl. Prob. 14, 811–832.Google Scholar
[2] Doney, R. A. (1987) On the maxima of random walks and stable processes and the arc-sine law. Bull. London. Math. Soc. 19, 177–182.CrossRefGoogle Scholar
[3] Doney, R. A. (in preparation) A path decomposition for Lévy processes.Google Scholar
[4] Doney, R. A. (1989) Last exit times for random walks. Stoch. Proc. Appl. Google Scholar
[5] El Karoui, N. and Chaleyat-Maurel, M. (1978) Un probléme de réflexion et ses applications au temps local et aux equations différentielles stochastiques sur R: cas continu. Astérisque 52–53, 117–144.Google Scholar
[6] Imhof, J-P. (1986) On the time spent above a level by Brownian motion with negative drift. Adv. Appl. Prob. 18, 1017–1018.Google Scholar
[7] Pitman, J. and Yor, M. (1986) Asymptotic laws of planar Brownian motion. Ann. Prob. 11, 733–779.Google Scholar
[8] Williams, D. (1974) Path decomposition and continuity of local time for one-dimensional diffusions, I. Proc. London Math. Soc. 28, 738–768.CrossRefGoogle Scholar