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A Clark-Ocone formula in UMD Banach spaces

   
@article{ECP1361,
	author = {Jan Maas and Jan Neerven},
	title = {A Clark-Ocone formula in UMD Banach spaces},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {13},
	year = {2008},
	keywords = {Clark-Ocone formula, Malliavin calculus},
	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.},
	pages = {no. 15, 151-164},
	issn = {1083-589X},
	doi = {10.1214/ECP.v13-1361},    
        url = {http://ecp.ejpecp.org/article/view/1361}}