On some Brownian functionals and their applications to moments in the lognormal stochastic volatility model

@article{JacekJakubowski2013,
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.},
author = {Jacek Jakubowski, Maciej Wiśniewolski},
journal = {Studia Mathematica},
keywords = {geometric Brownian motion; Ornstein-Uhlenbeck process; Laplace transform; Bessel process; transform},
language = {eng},
number = {3},
pages = {201-224},
title = {On some Brownian functionals and their applications to moments in the lognormal stochastic volatility model},
url = {http://eudml.org/doc/285917},
volume = {219},
year = {2013},
}

TY - JOUR
AU - Jacek Jakubowski
AU - Maciej Wiśniewolski
TI - On some Brownian functionals and their applications to moments in the lognormal stochastic volatility model
JO - Studia Mathematica
PY - 2013
VL - 219
IS - 3
SP - 201
EP - 224
AB - 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.
LA - eng
KW - geometric Brownian motion; Ornstein-Uhlenbeck process; Laplace transform; Bessel process; transform
UR - http://eudml.org/doc/285917
ER -