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Clark-Ocone type formulas on spaces of test and generalized functions of Meixner white noise analysis


Abstract

In the classical Gaussian analysis the Clark-Ocone formula can be written in the form $$ F=\mathbf EF+\int\mathbf E_t\partial_t FdW_t, $$ where the function (the random variable) $F$ is square integrable with respect to the Gaussian measure and differentiable by Hida; $\mathbf E$ denotes the expectation; $\mathbf E_t$ denotes the conditional expectation with respect to the full $\sigma$-algebra that is generated by a Wiener process $W$ up to the point of time $t$; $\partial_\cdot F$ is the Hida derivative of $F$; $\int\circ (t)dW_t$ denotes the It\^o stochastic integral with respect to the Wiener process. This formula has applications in the stochastic analysis and in the financial mathematics. In this paper we generalize the Clark-Ocone formula to spaces of test and generalized functions of the so-called Meixner white noise analysis, in which instead of the Gaussian measure one uses the so-called generalized Meixner measure $\mu$ (depending on parameters, $\mu$ can be the Gaussian, Poissonian, Gamma measure etc.). In particular, we study properties of integrands in our (Clark-Ocone type) formulas.


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Article Information

TitleClark-Ocone type formulas on spaces of test and generalized functions of Meixner white noise analysis
SourceMethods Funct. Anal. Topology, Vol. 18 (2012), no. 2, 160-175
MathSciNet MR2978192
zbMATH 1265.60127
CopyrightThe Author(s) 2012 (CC BY-SA)

Authors Information

N. A. Kachanovsky
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine 


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Citation Example

N. A. Kachanovsky, Clark-Ocone type formulas on spaces of test and generalized functions of Meixner white noise analysis, Methods Funct. Anal. Topology 18 (2012), no. 2, 160-175.


BibTex

@article {MFAT610,
    AUTHOR = {Kachanovsky, N. A.},
     TITLE = {Clark-Ocone type formulas on spaces of test and generalized functions of Meixner white noise analysis},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {18},
      YEAR = {2012},
    NUMBER = {2},
     PAGES = {160-175},
      ISSN = {1029-3531},
  MRNUMBER = {MR2978192},
 ZBLNUMBER = {1265.60127},
       URL = {http://mfat.imath.kiev.ua/article/?id=610},
}


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