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 Levy’s and Pitman’s theorems are discussed.
Highlights
Let B = {Bt, t 0} be a one-dimensional Brownian motion starting from 0 and defined on a probability space (Ω, F, P )
The purpose of this second part of our surveys is to present some results obtained by applying the formulae and identities mentioned in Part I to Brownian motion and some related stochastic processes
Gruet [28] has considered the Brownian motion on Hn, which is a diffusion process generated by ∆n/2, and has derived a new integral representation for pn(t, r) by using the explicit expression (1.2) for the joint density of (A(tμ), Bt(μ))
Summary
The purpose of this second part of our surveys is to present some results obtained by applying the formulae and identities mentioned in Part I to Brownian motion and some related stochastic processes. Dufresne’s relation is important in studying extensions or analogues of Levy’s and Pitman’s theorems about, respectively, {Mt(μ) − Bt(μ)} and {2Mt(μ) − Bt(μ)}, where Mt(μ) = max0 s t Bs(μ), by means of exponential functionals. The classical Levy and Pitman theorems may be seen as limiting results of those as λ → ∞
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