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Some asymptotic results for exponential functionals of Brownian motion

Part of: Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

J.-C. Gruet  and
Z. Shi
J.-C. Gruet*
Affiliation:
Université Paris VI
Z. Shi*
Affiliation:
Université Paris VI
*
∗Postal address: Laboratoire de Probabilités, CNRS URA 224, Université Paris VI, Tour 56, 4 Place Jussieu, 75252 Paris Cedex 05, France and Université de Reims Champagne Ardenne, UFR Sciences, B.P. 1039, 51687 Reims Cedex 2, France.
∗∗Postal address: L.S.T.A.-CNRS URA 1321, Université Paris VI, Tour 45–55, 3e étage, 4 Place Jussieu, 75252 Paris Cedex 05, France.
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Abstract

The study of exponential functionals of Brownian motion has recently attracted much attention, partly motivated by several problems in financial mathematics. Let be a linear Brownian motion starting from 0. Following Dufresne (1989), (1990), De Schepper and Goovaerts (1992) and De Schepper et al. (1992), we are interested in the process (for δ > 0), which stands for the discounted values of a continuous perpetuity payment. We characterize the upper class (in the sense of Paul Lévy) of X, as δ tends to zero, by an integral test. The law of the iterated logarithm is obtained as a straightforward consequence. The process exp(W(u))du is studied as well. The class of upper functions of Z is provided. An application to the lim inf behaviour of the winding clock of planar Brownian motion is presented.

Keywords

LAW OF THE ITERATED LOGARITHMLÉVY'S UPPER CLASSEXPONENTIAL FUNCTIONALWINDING CLOCKBROWNIAN MOTION

MSC classification

Primary: 60F15: Strong theorems
Secondary: 60J65: Brownian motion
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 4 , December 1995 , pp. 930 - 940
Copyright
Copyright © Applied Probability Trust 1995 

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