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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 

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