Some asymptotic results for exponential functionals of Brownian motion

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.