Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T08:25:13.641Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
∗∗Postal address: Laboratoire de Probabilitiés, Université Paris VI, Tour 56, 4 Place Jussieu, 75252 Paris, France.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Akahori, J. (1995). Some formulae for a new type of path-dependent option. Ann. Appl. Prob. 5, 383–388.Google Scholar
Azéma, J., and Yor, M. (1989). Etude d'une martingale remarquable. Sem. de Prob. XXIII, Lect. Notes in Maths. 1372, 88–130.Google Scholar
Bertoin, J. (1991). Décomposition du mouvement brownien avec dérivé en un minimum local par juxtaposition de ses excursions positives et negatives. Sem. de Prob. XXV, Lect. Notes in Maths. 1485, 330–344.CrossRefGoogle Scholar
Bertoin, J., Chaumont, L., and Yor, M. (1997). Two chain-transformations and their applications to quantiles. J. Appl. Prob. 34, 882–897.Google Scholar
Biane, Ph., and Yor, M. (1988). Quelques précisions sur le meandre brownien. Bull. Sci. Maths, 2nd Série 112, 101–109.Google Scholar
Brandt, A. (1961). A generalization of a combinatorial theorem of Sparre Andersen about sums of random variables. Math. Scand. 9, 352–358.CrossRefGoogle Scholar
Chung, K.L. (1976). Excursions in Brownian motion. Ark. Mat. 14, 155–177.Google Scholar
Csaki, E., Földes, A., and Salminen, P. (1987). On the joint distribution of the maximum and its location for a linear diffusion. Ann. Inst. Poincaré, Section B 23, 178–194.Google Scholar
Denisov, I.V. (1985). A random walk and a Wiener process near a maximum. Th. Prob. Appl. 713–716.Google Scholar
Embrechts, P., Rogers, L.C. G., and Yor, M. (1995). A proof of Dassios' representation of the α-quantile of Brownian Motion with drift. Ann. Appl. Prob. 5, 757–767.CrossRefGoogle Scholar
Hobby, C., and Pyke, R. (1963). Remarks on the equivalence principle in fluctuation theory. Math. Scand. 12, 19–24.Google Scholar
Imhof, J.-P. (1984). Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Prob. 21, 500–510.CrossRefGoogle Scholar
Imhof, J.-P. and Kümmerling, P. (1986). Operational derivation of some Brownian motion results. Internat. Statistical Rev. 54, 327–341.Google Scholar
Karatzas, I., and Shreve, M. (1987). Brownian Motion and Stochastic Calculus. Springer, Berlin.Google Scholar
Karatzas, I., and Shreve, M. (1987). A decomposition of the Brownian path. Statist. Prob. Let. 5, 87–94.Google Scholar
Perkins, E. (1982). Local time is a semi-martingale. Z. Wahrscheinlichtkeitsth. 60, 79–117.Google Scholar
Revuz, D., and Yor, M. (1994). Continuous Martingales and Brownian Motion. 2nd edn. Springer.Google Scholar
Takács, L. (1995). On a generalisation of the Arc-sine law. Ann. Appl. Prob. 6, 1035–1040.Google Scholar
Yor, M. (1995). The distribution of Brownian quantiles. J. Appl. Prob. 32, 405–416.Google Scholar