Electronic Communications in Probability

A Clark-Ocone formula in UMD Banach spaces

Jan Maas and Jan Neerven

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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.

Article information

Source
Electron. Commun. Probab., Volume 13 (2008), paper no. 15, 151-164.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465233442

Digital Object Identifier
doi:10.1214/ECP.v13-1361

Mathematical Reviews number (MathSciNet)
MR2399277

Zentralblatt MATH identifier
1189.60111

Subjects
Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus
Secondary: 46B09: Probabilistic methods in Banach space theory [See also 60Bxx] 60H05: Stochastic integrals

Keywords
Clark-Ocone formula Malliavin calculus

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Maas, Jan; Neerven, Jan. A Clark-Ocone formula in UMD Banach spaces. Electron. Commun. Probab. 13 (2008), paper no. 15, 151--164. doi:10.1214/ECP.v13-1361. https://projecteuclid.org/euclid.ecp/1465233442


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