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The Integral of the Supremum Process of Brownian Motion

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Svante Janson  and
Niclas Petersson
Svante Janson*
Affiliation:
Uppsala University
Niclas Petersson*
Affiliation:
Uppsala University
*
∗Postal address: Department of Mathematics, Uppsala University, PO Box 480, SE–751 06 Uppsala, Sweden.
∗Postal address: Department of Mathematics, Uppsala University, PO Box 480, SE–751 06 Uppsala, Sweden.
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Abstract

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In this paper we study the integral of the supremum process of standard Brownian motion. We present an explicit formula for the moments of the integral (or area) (T) covered by the process in the time interval [0,T]. The Laplace transform of (T) follows as a consequence. The main proof involves a double Laplace transform of (T) and is based on excursion theory and local time for Brownian motion.

Keywords

Brownian motionsupremum processlocal timeBrownian areas

MSC classification

Secondary: 60J65: Brownian motion 60J55: Local time and additive functionals
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 2 , June 2009 , pp. 593 - 600
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Davies, L. (1976). Tail probabilities for positive random variables with entire characteristic functions of very regular growth. Z. Angew. Math. Mech. 56, 334–336.CrossRefGoogle Scholar
[2] Fill, J. A. and Janson, S. (2009). Precise logarithmic asymptotics for the right tails of some limit random variables for random trees. Ann. Combinatorics 12, 403–416.Google Scholar
[3] Janson, S. (2007). Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas. Prob. Surveys 4, 80–145.Google Scholar
[4] Janson, S. and Chassaing, P. (2004). The center of mass of the ISE and the Wiener index of trees. Electron. Commun. Prob. 9, 178–187.Google Scholar
[5] Janson, S. and Louchard, G. (2007). Tail estimates for the Brownian excursion area and other Brownian areas. Electron. J. Prob. 12, 1600–1632.Google Scholar
[6] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.Google Scholar
[7] Kasahara, Y. (1978). Tauberian theorems of exponential type. J. Math. Kyoto Univ. 18, 209–219.Google Scholar
[8] Perman, M. and Wellner, J. A. (1996). On the distribution of Brownian areas. Ann. Appl. Prob. 6, 1091–1111.Google Scholar
[9] Petersson, N. (2008). The maximum displacement for linear probing hashing. Preprint, U.U.D.M. Rep. 2008:6, Uppsala University.Google Scholar
[10] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin.CrossRefGoogle Scholar