Generalized covariation for Banach space valued processes, Itô formula and applications

This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related Itô formula. If $\X$ and $\Y$ take respectively values in Banach spaces $B_{1}$ and $B_{2}$ and $χ$ is a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$ (denoted by $(B_{1}\hat{\otimes}_πB_{2})^{\ast}$), we define the so-called $χ$-covariation of $\X$ and $\Y$. If $\X=\Y$, the $χ$-covariation is called $χ$-quadratic variation. The notion of $χ$-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 $χ$ is the whole space $(B_{1}\hat{\otimes}_πB_{1})^{\ast}$ then the $χ$-quadratic variation coincides with the quadratic variation of a $B_{1}$-valued semimartingale. We evaluate the $χ$-covariation of various processes for several examples of $χ$ with a particular attention to the case $B_{1}=B_{2}=C([-τ,0])$ for some $τ>0$ and $\X$ and $\Y$ being \textit{window processes}. If $X$ is a real valued process, we call window process associated with $X$ the $C([-τ,0])$-valued process $\X:=X(\cdot)$ defined by $X_t(y) = X_{t+y}$, where $y \in [-τ,0]$. The Itô 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 $h=H(X_{T}(\cdot))$, $H:C([-T,0])\longrightarrow\R$ for not-necessarily semimartingales $X$ with finite quadratic variation. This representation will be linked to a function $u:[0,T]\times C([-T,0])\longrightarrow \mathbb{R}$ solving an infinite dimensional partial differential equation.

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