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 B1 and B2 and ! is a suitable subspace of the dual of the projective tensor product of B1 and B2, we define the so-called !-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 dual of (B1 tensor B1) then the chi-quadratic variation coincides with the quadratic variation of a B1-valued semimartingale. We evaluate the chi-covariation of various processes for several examples of ! with a particular attention to the case B1 = B2 = C([−T, 0]) for some T> 0 and X and Y being window processes. If X is a real valued process, we call window process associated with X the C([−R, 0])-valued process X := X(·) defined by Xt(y) = Xt+y, where y in [−T, 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(XT (·)), with H defined on C([−T, 0]) to R for not-necessarily semimartingales X with finite quadratic variation. This representation will be linked to a function u : [0, T]×C([−T, 0]) to R solving an infinite dimensional partial differential equation.