A Clark-Ocone formula in UMD Banach spaces
Abstract
Let H be a separable real Hilbert space and let F = (Ft)t∈ [0,T] be the augmented filtration generated by an H-cylindrical Brownian motion WH on [0,T]. We prove that if E is a UMD Banach space, 1≤ p<∞, and f∈ D1,p(E) is FT-measurable, then f = f + ∫0T PF(Df) dWH where D is the Malliavin derivative and PF 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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