Some Brownian functionals and their laws

On Some Exponential Functionals of Brownian Motion

In this paper, distributional questions which arise in certain Mathematical Finance models are studied: the distribution of the integral over a fixed time interval {0, T} of the exponential of Brownian motion with drift is computed explicitly, with the help of former computations made by the author for Bessel processes. The moments of this integral are obtained independently and take a particul...

Exponential functionals of Brownian motion, II: Some related diffusion processes

This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and ...

Some limiting laws associated with the integrated Brownian motion

We study some limit theorems for the normalized law of integrated Brownian motion perturbed by several examples of functionals: the first passage time, the n passage time, the last passage time up to a finite horizon and the supremum. We show that the penalization principle holds in all these cases and give descriptions of the conditioned processes. In particular, it is remarkable that the pena...

Analysis of Complex Brownian Functionals

Exponential functionals of Brownian motion, I: Probability laws at fixed time

This paper is the first part of our survey on various results about the distribution of exponential type Brownian functionals defined as an integral over time of geometric Brownian motion. Several related topics are also mentioned.