Generalized covariation for Banach space valued processes and Itô formula

june, 2014
Publication type:
Paper in peer-reviewed journals
Journal:
Osaka Journal of Mathematics, vol. 51(3)
arXiv:
images/icons/icon_arxiv.png 1105.4419
Keywords :
Covariation and Quadratic variation, Calculus via regularization, Infinite dimensional analysis, Tensor analysis, Itô formula, Stochastic integration, Fractional Brownian motion, Dirichlet processes.
Abstract:
This paper concerns the notion of quadratic variation and covariation for Banach valued processes and related Itô formula. If $\X$ and $\Y$ take respectively values in Banach spaces $B_{1}$ and $B_{2}$ (denoted by $(B_{1}\hat{\otimes}_{\pi}B_{2})^{\ast}$) and $\chi$ is a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$ we define the so-called $\chi$-covariation of $\X$ and $\Y$. If $\X=\Y$ the $\chi$-covariation is called $\chi$-quadratic variation. The notion of $\chi$-quadratic variation is a natural generalization of the one introduced by Métivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if $\chi$ is the whole space $(B_{1}\hat{\otimes}_{\pi}B_{1})^{\ast}$ then the $\chi$-quadratic variation coincides with the quadratic variation of a $B_{1}$-valued semimartingale. We evaluate the $\chi$-covariation of various processes for several examples of $\chi$ with a particular attention to the case $B_{1}=B_{2}=C([-\tau,0])$ for some $\tau>0$ and $\X$ and $\Y$ being \textit{window processes}. If $X$ is a real process, we call window process associated with $X$ the $C([-\tau,0])$-valued process $\X:=X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-\tau,0]$.
BibTeX:
@article{DiG-Rus-2014,
    author={Cristina Di Girolami and Francesco Russo },
    title={Generalized covariation for Banach space valued processes and 
           Itô formula },
    journal={Osaka Journal of Mathematics },
    year={2014 },
    month={6},
    volume={51(3) },
}