A Clark-Ocone formula in UMD Banach spaces

Details A Clark-Ocone formula in UMD Banach spaces A Clark-Ocone formula in UMD Banach spaces

2007-09-13

arxiv.org/abs/0709.2021v2

Mathematics

Probability

12 pages; revised version, to appear in Electronic Communications in Probability

Scientific paper

Let H be a separable real Hilbert space and let F = (F_t)_{t\in [0,T]} be the augmented filtration generated by an H-cylindrical Brownian motion W_H on [0,T]. We prove that if E is a UMD Banach space, 1\leq p<\infty, and f\in D^{1,p}(E) is F_T-measurable, then f = \E f + \int_0^T P_F(Df) dW_H where D is the Malliavin derivative and P_F is the projection onto the F-adapted elements in a suitable Banach space of L^p-stochastically integrable L(H,E)-valued processes.

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