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2008 A Clark-Ocone formula in UMD Banach spaces
Jan Maas, Jan Neerven
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Electron. Commun. Probab. 13: 151-164 (2008). DOI: 10.1214/ECP.v13-1361


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.


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Jan Maas. Jan Neerven. "A Clark-Ocone formula in UMD Banach spaces." Electron. Commun. Probab. 13 151 - 164, 2008.


Accepted: 7 April 2008; Published: 2008
First available in Project Euclid: 6 June 2016

zbMATH: 1189.60111
MathSciNet: MR2399277
Digital Object Identifier: 10.1214/ECP.v13-1361

Primary: 60H07
Secondary: 46B09 , 60H05

Keywords: Clark-Ocone formula , Malliavin calculus

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