Clark-Ocone type formulas on the spaces of nonregular generalized functions in the Lévy white noise analysis
Abstract
In the classical Gaussian analysis the Clark-Ocone formula can be written in the form
\[
F=\mathbf EF+\int\mathbf E\big(\partial_t F|_{\mathcal F_t}\big)dW_t,
\]
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
$\sigma$-algebra $\mathcal F_t$ 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 Itô stochastic integral over a 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 the spaces $(\mathcal H_{-\tau})_{-q}$ of
nonregular generalized functions in the Lévy white noise analysis. More exactly, we prove
that any element of $(\mathcal H_{-\tau})_{-q}$ can be represented as a sum of its expectation
and a result of stochastic integration over a Lévy process of some generalized function, and
construct Clark-Ocone type formulas on $(\mathcal H_{-\tau})_{-q}$ and on its subsets.
Key words: Lévy process, extended stochastic integral, stochastic derivative, Clark-Ocone formula.
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