Clark-Ocone type formula for non-semimartingales with finite quadratic variation

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We provide a suitable framework for the concept of finite quadratic variation for processes with values in a separable Banach space B using the language of stochastic calculus via regularizations, introduced in the case B= by the second author and P. Vallois. To a real continuous process X we associate the Banach valued process X(·), called window process, which describes the evolution of X taking into account a memory τ>0. The natural state space for X(·) is the Banach space of continuous functions on [-τ,0]. If X is a real finite quadratic variation process, an appropriated Itô formula is presented, from which we derive a generalized Clark-Ocone formula for non-semimartingales having the same quadratic variation as Brownian motion. The representation is based on solutions of an infinite dimensional PDE.