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

Tom 219 / 2013

Jacek Jakubowski, Maciej Wiśniewolski Studia Mathematica 219 (2013), 201-224 MSC: 60J65, 60J60, 91B25, 91G80, 44A10. DOI: 10.4064/sm219-3-2

Streszczenie

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.

Autorzy

  • Jacek JakubowskiInstitute of Mathematics
    Warsaw University
    02-097 Warszawa, Poland
    e-mail
  • Maciej WiśniewolskiInstitute of Mathematics
    Warsaw University
    02-097 Warszawa, Poland
    e-mail

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