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A Clark-Ocone formula in UMD Banach spaces


 
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1. Title Title of document A Clark-Ocone formula in UMD Banach spaces
 
2. Creator Author's name, affiliation, country Jan Maas; TU Delft
 
2. Creator Author's name, affiliation, country Jan van Neerven; TU Delft
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Clark-Ocone formula, Malliavin calculus
 
3. Subject Subject classification Primary: 60H07; Secondary: 46B09, 60H05
 
4. Description Abstract Let $H$ be a separable real Hilbert space and let $\mathbb{F}=(\mathscr{F}_t)_{t\in [0,T]}$ be the augmented filtration generated by an $H$-cylindrical Brownian motion $(W_H(t))_{t\in [0,T]}$ on a probability space $(\Omega,\mathscr{F},\mathbb{P})$. We prove that if $E$ is a UMD Banach space, $1\le p<\infty$, and $F\in \mathbb{D}^{1,p}(\Omega;E)$ is $\mathscr{F}_T$-measurable, then $$ F = \mathbb{E} (F) + \int_0^T P_{\mathbb{F}} (DF)\,dW_H,$$ where $D$ is the Malliavin derivative of $F$ and $P_{\mathbb{F}}$ is the projection onto the ${\mathbb{F}}$-adapted elements in a suitable Banach space of $L^p$-stochastically integrable $\mathscr{L}(H,E)$-valued processes.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) Netherlands Organisation for Scientific Research (NWO), Australian Research Council (ARC).
 
7. Date (YYYY-MM-DD) 2008-04-07
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1361
 
10. Identifier Digital Object Identifier 10.1214/ECP.v13-1361
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 13
 
12. Language English=en
 
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