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 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 (B1 ⊗π B1) * then the χ-quadratic variation coincides with the quadratic variation of a B1-valued semimartingale. We evaluate the χ-covariation of various processes for several examples of χ with a particular attention to the case B1 = B2 = 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 Xt(y) = Xt+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(XT (•)), 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.

This paper concerns a class of Banach valued processes which have finite quadratic variation. The notion introduced here generalizes the classical one, of Métivier and Pellaumail which is quite restrictive. We make use of the notion of χ-covariation which is a generalized notion of covariation for processes with values in two Banach spaces B1 and B2. χ refers to a suitable subspace of the dual of the projective tensor product of B1 and B2. We investigate some C 1 type transformations for various classes of stochastic processes admitting a χ-quadratic variation and related properties. If X 1 and X 2 admit a χ-covariation, F i : Bi → R, i = 1, 2 are of class C 1 with some supplementary assumptions then the covariation of the real processes F 1 (X 1 ) and F 2 (X 2 ) exist. A detailed analysis will be devoted to the so-called window processes. Let X be a real continuous process; the C([−τ, 0])-valued process X(·) defined by Xt(y) = Xt+y, where y ∈ [−τ, 0], is called window process. Special attention is given to transformations of window processes associated with Dirichlet and weak Dirichlet processes. In fact we aim to generalize the following properties valid for B = R. If X = X is a real valued Dirichlet process and F : B → R of class C 1 (B) then F (X) is still a Dirichlet process. If X = X is a weak Dirichlet process with finite quadratic variation, and F : C 0,1 ([0, T ] × B) is of class C 0,1 , then (F (t, Xt)) is a weak Dirichlet process. We specify corresponding results when B = C([−τ, 0]) and X = X(·). This will consitute a significant Fukushima decomposition for functionals of windows of (weak) Dirichlet processes. As applications, we give a new technique for representing path-dependent random variables.[2010 Math Subject Classification: ] 60H05, 60H07, 60H10, 60H30, 91G80We say that X and Y admit a covariation ift exists in probability for every t > 0 and ii) the limiting process in i) admits a continuous modification that will be denoted by [X, Y ].If [X, X] exists, we say that X is a finite quadratic variation process (or it has finite quadratic variation) and it is also denoted by [X]. If [X] = 0, X is called zero quadratic variation process. We say that (X, Y ) admits its (or X and Y admit their) mutual covariations if [X], [Y ] and [X, Y ] exist.Remark 1.2.1. Lemma 1.3 below allows to show that, whenever [X, X] exists, then [X, X] ε also converges in the ucp sense as intended for instance in the [23, 25] sense. The basic results established there are still valid here, see the following items.2. If X and Y are (F t )-local semimartingales, then [X, Y ] coincides with the classical covariation, see Corollaries 2 and 3 in [25].3. If X (resp. A) is a finite (resp. zero) quadratic variation process, we have [A, X] = 0.We recall two useful tools related to the notion of covariation for real valued processes, see Lemma 3.1 from [24] and Propostion 1.2 in [23].Lemma 1.3. Let (Z ǫ ) ǫ>0 be a family of continuous processes indexed by [0, T ]. We suppose the following. 1) ∀ǫ > 0, t −→ Z ǫ t is increasing. 2) Th...