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 χ is a suitable subspace of the dual of the projective tensor product of B₁ and B₂ (denoted by (B₁ ⊗_π B₂)*), we define the so-called X-covariation of X and Y. If X = Y, the X-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 X is the whole space (B1 ⊗ݰπ B1)* then the χ-quadratic variation coincides with the quadratic variation of a B1-valued semimartingale. We evaluate the τ -covariation of various processes for several examples of X with a particular attention to the case B1 D B2 D 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 W [0, T] x C([―T, 0]) → ℝ solving an infinite dimensional partial differential equation.