Clark-Ocone type formulas in the Meixner white noise analysis

Abstract

In the classical Gaussian analysis the Clark-Ocone formula allows to reconstruct an integrand if we know the Itô stochastic integral. This formula can be written in the form $$ F=\mathbf EF+\int\mathbf E\big\{\partial_t F|_{\mathcal F_t}\big\} dW_t, $$ where a function (a random variable) $F$ is square integrable with respect to the Gaussian measure and differentiable by Hida; $\mathbf E$ $-$ the expectation; $\mathbf E\big\{\circ|_{\mathcal F_t}\big\}$ $-$ the conditional expectation with respect to a full $\sigma$-algebra $\mathcal F_t$ that is generated by the Wiener process $W$ up to the point of time $t$; $\partial_\cdot F$ $-$ the Hida derivative of $F$; $\int\circ (t)dW_t$ $-$ the Itô stochastic integral with respect to the Wiener process.

In this paper, we explain how to reconstruct an integrand in the case when instead of the Gaussian measure one considers the so-called generalized Meixner measure $\mu$ (depending on parameters, $\mu$ can be the Gaussian, Poissonian, Gamma measure etc.) and obtain corresponding Clark-Ocone type formulas.