Generalized covariation for Banach space valued processes, It\^o formula and applications

This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related It^o formula. If $\text{X}$ and $\text{Y}$ take respectively values in Banach spaces $B_1$ and $B_2$ and $\chi$ is a suitable subspace of the dual of the projective tensor product of $B_1$ and $B_2$ (denoted by $(B_1{\widehat{\otimes }}_{\pi }{B}_2{)}^{\ast }$), we define the so-called $\chi$-covariation of $\text{X}$ and $\text{Y}$. If $\text{X}=\text{Y}$, the $\chi$-covariation is called $\chi$-quadratic variation. The notion of $\chi$-quadratic variation is a natural generalization of the one introduced by M'etivier-Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if $\chi$ is the whole space $(B_1{\widehat{\otimes }}_{\pi }{B}_1{)}^{\ast }$ 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 $\text{X}$ and $\text{Y}$ being $\mathit{\text{window}}\mathit{\text{processes}}$. If $X$ is a real valued process, we call window process associated with $X$ the $C([-\tau ,0])$-valued process $\text{X}:=X(\cdot )$ defined by $X_t(y)=X_{t+y}$, where $y\in [-\tau ,0]$. The It^o formula introduced here is an important instrument to establish a representation result of Clark-Ocone type for a class of path dependent random variables of type $h=H(X_T(\cdot ))$, $H:C([-T,0])⟶\mathbb{R}$ for not-necessarily semimartingales $X$ with finite quadratic variation. This representation will be linked to a function $u:[0,T]\times C([-T,0])⟶\mathbb{R}$ solving an infinite dimensional partial differential equation.