Malliavin calculus on extensions of abstract Wiener spaces

Abstract

Malliavin calculus is developed in a uniform way for (possibly non separable) extensions of $L^p(W_{C_{\mathbb{F}}})$, where $W_{C_{\mathbb{F}}}$ is the Wiener measure on the space $C_{\mathbb{F}}$ of continuous functions from $[0,1]$ into any abstract Wiener Fréchet space $\mathbb{F}$ over a fixed separable Hilbert space $\mathbb{H}$. Since the continuous time line is available in $C_{\mathbb{F}}$ , we can prove the Clark- Ocone formula for these extensions, we study time-anticipating Girsanov transformations and prove that Skorohod integral processes for finite chaos levels have continuous modifications. We use a rich probability space with measure $\widehat{\Gamma }_{\mathbb{H}}$,which only depends on $\mathbb{H}$, such that for any $p\in [0,\infty [$, $ L^p\left( W_{C_{\mathbb{F}}}\right) $ can be canonically embedded into $L^p\left( \widehat{\Gamma }_{\mathbb{H}}\right) $ for any abstract Wiener Fréchet space $\mathbb{F}$ over $\mathbb{H}$.