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

Abstract

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 = R 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.