On the Clark Ocone formula for the abstract Wiener space

Fix an abstract Wiener space (H, B) where H is a separable Hilbert space densely embedded into a Banach space B. A pathwise construction of the Ito integral as a continuous square integrable martingale is given, where the integrands are H-valued processes and the integrator is a B-valued Brownian motion. We use this approach to the vector integral to prove that each Malliavin differentiable functional phi defined on the space C-B of continuous B-valued functions on [0, 1], endowed with the Wiener measure, can be decomposed into the sum of the expected value of phi and the Ito integral of the conditional expectation of the Malliavin derivative of phi with respect to the Brownian filtration. The Malliavin derivative of phi is an H-valued stochastic process. In a second application, it is shown that the iterated Ito integral, defined as a process on C-B x [0, 1], is a continuous square integrable martingale. Published by Elsevier Science (USA).