Generalized covariation for Banach space valued processes and Itô formula

june, 2014
Publication type:
Paper in peer-reviewed journals
Osaka Journal of Mathematics, vol. 51(3)
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.
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 $y \in [-\tau,0]$.
    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 },
    volume={51(3) },