Abstract: 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 extensions of Lévy’s and Pitman’s theorems are discussed.

This is survey of some recent results connecting random matrices, noncolliding processes and queues.

www.elsevier.com/locate/spa Conditioned stochastic di erential equations: theory, examples and application to nance

Diffusion processes, Exponential analogue of the 2M − X Pitman’s theorem We present a class of processes which enjoy an exponential analogue of Pitman’s 2M-X theorem, improving hence some works of H. Matsumoto and M. Yor. 1