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
arXiv:
images/icons/icon_arxiv.png 1005.3608
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},
}