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

Abstract

This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related Itô formula. If $\mathbb{X}$ and $\mathbb{Y}$ 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 $(B_{1} \hatotimes_{\pi}B_{2})^{*}$), we define the so-called $\chi$-covariation of $\mathbb{X}$ and $\mathbb{Y}$. If $\mathbb{X} = \mathbb{Y}$, 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 $(B_{1} \hatotimes_{\pi}B_{1})^{*}$ 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 $\mathbb{X}$ and $\mathbb{Y}$ being window processes. If $X$ is a real valued process, we call window process associated with $X$ the $C([-\tau, 0])$-valued process $\mathbb{X} := X({}\cdot{})$ defined by $X_{t}(y) = X_{t+y}$, where $y \in [-\tau, 0]$. 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 $h = H(X_{T}({}\cdot{}))$, $H\colon C([-T, 0])\to\mathbb{R}$ for not-necessarily semimartingales $X$ with finite quadratic variation. This representation will be linked to a function $u\colon [0, T]\times C([-T, 0])\to \mathbb{R}$ solving an infinite dimensional partial differential equation.