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 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 ⊗^^<^>_<π> B_2)^<*>), 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 ⊗^^<^>_<π> B_2)^<*> then the -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 window processes. If X is a real valued process, we call window process associated with X the C([-τ, 0])-valued process X = X(・) defined by X_t(y) = X_<t+y> , where y ∈ [-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(・)), H : C([-T,0])→R for not-necessarily semimartingales X with finite quadratic variation. This representation will be linked to a function u : [0, T] × C([-T, 0]) → R solving an infinite dimensional partial differential equation.