Journal of Mathematics of Kyoto University

Malliavin calculus on extensions of abstract Wiener spaces

Horst Osswald

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 $\mathbb{H}$.

Article information

Source
J. Math. Kyoto Univ., Volume 48, Number 2 (2008), 239-263.

Dates
First available in Project Euclid: 14 August 2009

https://projecteuclid.org/euclid.kjm/1250271411

Digital Object Identifier
doi:10.1215/kjm/1250271411

Mathematical Reviews number (MathSciNet)
MR2436736

Zentralblatt MATH identifier
1181.60079

Citation

Osswald, Horst. Malliavin calculus on extensions of abstract Wiener spaces. J. Math. Kyoto Univ. 48 (2008), no. 2, 239--263. doi:10.1215/kjm/1250271411. https://projecteuclid.org/euclid.kjm/1250271411