Clark-Ocone type formulas on the spaces of regular basic and generalized functions in the analysis of Levy white noise

Abstract

In the classical Gaussian analysis the Clark-Ocone formula can be written in the form $$ F=\mathbf{E}{F}+\int\mathbf{E}\big(\partial_t F|_{\mathcal F_t}\big)\,dW_t, $$ where a function (a random variable) $F$ is square integrable with respect to the Gaussian measure and differentiable by Hida; $\mathbf{E}$ denotes the expectation; $\mathbf{E}\big(\circ|_{\mathcal F_t}\big)$---the conditional expectation with respect to the full $\sigma$-algebra $\mathcal F_t$ that is generated by the 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 many applications, in particular, in the stochastic analysis and in the financial mathematics.
In this paper we generalize the Clark-Ocone formula to spaces of regular test and generalized functions of the Levy white noise analysis. More exactly, we obtain different Clark-Ocone type formulas on the above-mentioned spaces, study the properties of the integrands in these formulas, establish the conditions under which a Clark-Ocone type formula takes a classical form, etc. In particular, we show that the restrictive condition of differentiability by Hida for a random variable is not really significant.