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On a formula of Takács for Brownian motion with drift

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

R. A. Doney  and
M. Yor
R. A. Doney*
Affiliation:
University of Manchester
M. Yor*
Affiliation:
Université Paris VI
*
∗Postal address: Statistical Laboratory, Department of Mathematics, University of Manchester, Oxford Road, Manchester, M13 9PL, UK. E-mail address: rad@ma.man.ac.uk
∗∗Postal address: Laboratoire de Probabilitiés, Université Paris VI, Tour 56, 4 Place Jussieu, 75252 Paris, France.
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Abstract

A recent result of Takács (1995) gives explicitly the density of the time spent before t above a level x ≠ 0 by Brownian motion with drift. Takács' proof is by means of random walk approximations to Brownian motion, but in this paper we give two different proofs of this result by considerations involving only Brownian motion. We also give a reformulation of Takács' result which involves Brownian meanders, and an extension of Denisov's representation of Brownian motion in terms of two independent Brownian meanders.

Keywords

Brownian motion with driftoccupation timesBrownian meanders

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 2 , June 1998 , pp. 272 - 280
Copyright
Copyright © Applied Probability Trust 1998 

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