On some Brownian functionals and their applications to moments in the lognormal stochastic volatility model
Volume 219 / 2013
Abstract
We find a probabilistic representation of the Laplace transform of some special functional of geometric Brownian motion using squared Bessel and radial Ornstein–Uhlenbeck processes. Knowing the transition density functions of these processes, we obtain closed formulas for certain expectations of the relevant functional. Among other things we compute the Laplace transform of the exponent of the $T$ transforms of Brownian motion with drift used by Donati-Martin, Matsumoto, and Yor in a variety of identities of duality type between functionals of Brownian motion. We also present links between geometric Brownian motion and Markov processes studied by Matsumoto and Yor. These results have wide applications. As an example of their use in financial mathematics we find the moments of processes representing the asset price in the lognormal volatility model.
