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 paper studies the law of any power of the integral of geometric Brownian motion over any finite time interval. As its main results, two integral representations for this law are derived. This is by enhancing the Laplace transform ansatz of [Y] with complex analytic methods, which is the main methodological contribution of the paper. The one of our integrals has a similar structure to that obtained in [Y], while the other is in terms of Hermite functions as those of [Du01]. Performing or not performing a certain Girsanov transformation is identified as the source of these two forms of the laws. For exponents equal to 1 our results specialize to those obtained in [Y], but for exponents equal to minus 1 they give representations for the laws which are markedly different from those obtained in [Du01].

We derive an integral representation for the fundamental solution of the Kolmogorov forward equation: ft = −((1+µx)f)x + (ν x 2 f)xx associated with the Shiryaev process X solving: dXt = (1+µXt) dt + σXt dBt where µ ∈ IR, ν = σ 2 /2> 0 and B is a standard Brownian motion. The method of proof is based upon deriving and inverting a Laplace transform. Basic properties of X needed in the proof are reviewed. 1.

The Wiener disorder problem seeks to determine a stopping time which is as close as possible to the (unknown) time of ’disorder ’ when the drift of an observed Wiener process changes from one value to another. In this paper we present a solution of the Wiener disorder problem when the horizon is finite. The method of proof is based on reducing the initial problem to a parabolic free-boundary problem where the continuation region is determined by a continuous curved boundary. By means of the change-of-variable formula containing the local time of a diffusion process on curves we show that the optimal boundary can be characterized as a unique solution of the nonlinear integral equation. 1.

We prove a formula conjectured in [14] for the free energy density of a directed polymer in a Brownian environment in 1 + 1 dimensions. Mathematics Subject Classification (2000): 60K37,82D30,60K25,60J65 1

A model of returns for the post-credit-crunch