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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.

### Article Information

 Title Clark-Ocone type formulas on spaces of test and generalized functions of Meixner white noise analysis Source Methods Funct. Anal. Topology, Vol. 18 (2012), no. 2, 160-175 MathSciNet MR2978192 zbMATH 1265.60127 Copyright The 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

### 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},
}