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

january, 2011

Publication type:

Paper in peer-reviewed journals

Journal:

Comptes Rendus de l'Académie des Sciences., vol. 349(3-4), pp. 209 214

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Keywords :

Calculus via regularization Infinite dimensional analysis Clark-Ocone formula Dirichlet processes Itô formula Quadratic variation Hedging theory without semimartingales.

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.

BibTeX:

@article{DiG-Rus-2011, author={Cristina Di Girolami and Francesco Russo }, title={Clark-Ocone type formula for non-semimartingales with finite quadratic variation }, doi={10.1016/j.crma.2010.11.032 }, journal={Comptes Rendus de l'Académie des Sciences. }, year={2011 }, month={1}, volume={349(3-4) }, pages={209 214}, }