Geodesic
On some Brownian functionals and their applications to moments in the lognormal stochastic volatility model
Jacek Jakubowski1 ; Maciej Wiśniewolski1
1 Institute of Mathematics Warsaw University 02-097 Warszawa, Poland
Studia Mathematica, Tome 219 (2013) no. 3, pp. 201-224

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

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.
DOI : 10.4064/sm219-3-2
Keywords: probabilistic representation laplace transform special functional geometric brownian motion using squared bessel radial ornstein uhlenbeck processes knowing transition density functions these processes obtain closed formulas certain expectations relevant functional among other things compute laplace transform exponent transforms brownian motion drift donati martin matsumoto yor variety identities duality type between functionals brownian motion present links between geometric brownian motion markov processes studied matsumoto yor these results have wide applications example their financial mathematics moments processes representing asset price lognormal volatility model

Jacek Jakubowski 1 ; Maciej Wiśniewolski 1

1 Institute of Mathematics Warsaw University 02-097 Warszawa, Poland 
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Jacek Jakubowski; Maciej Wiśniewolski. On some Brownian functionals and their applications to moments in the lognormal stochastic volatility model. Studia Mathematica, Tome 219 (2013) no. 3, pp. 201-224. doi: 10.4064/sm219-3-2

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