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.

Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of real-valued Lévy processes ξ = (ξt, t ≥ 0). 0

We provide an integral formula for the Poisson kernel of half-spaces for Brownian motion in real hyperbolic space H n. This enables us to find asymptotic properties of the kernel. Our starting point is the formula for its Fourier transform. When n = 3, 4 or 6 we give an explicit formula for the Poisson kernel itself. In the general case we give various asymptotics and show convergence to the Poisson kernel of H n.