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

Cristina Di Girolami 1, 2 Francesco Russo 2, 3
3 MATHFI - Financial mathematics
Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech, UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12
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 $X(\cdot)$, called {\it window} process, which describes the evolution of $X$ taking into account a memory $\tau>0$. The natural state space for $X(\cdot)$ is the Banach space of continuous functions on $[-\tau,0]$. 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.
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Cristina Di Girolami, Francesco Russo. Clark-Ocone type formula for non-semimartingales with finite quadratic variation. Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2011, 349 (3-4), pp.209-214. ⟨10.1016/j.crma.2010.11.032⟩. ⟨inria-00484993v2⟩

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