A Clark-Ocone formula in UMD Banach spaces
Jan Maas, TU Delft
Jan van Neerven, TU Delft
Abstract
Let H be a separable real Hilbert space and let F = (Ft)t in [0,T] be the augmented filtration generated by an H-cylindrical Brownian motion WH on [0,T] on a probability space (Ω,F,P). We prove that if E is a UMD Banach space, 1≤ p<∞, and f in D1,p(E) is FT-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 Lp-stochastically integrable L(H,E)-valued processes.
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