Generalized covariation and extended Fukushima decompositions for Banach valued processes. Application to windows of Dirichlet processes

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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 C¹ type transformations for various classes of stochastic processes admitting a χ-quadratic variation and related properties. If ¹ and ² admit a χ-covariation, Fⁱ: Bᵢ →, i = 1, 2 are of class C¹ with some supplementary assumptions then the covariation of the real processes F¹(¹) and F²(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=. If =X is a real valued Dirichlet process and F:B → of class C¹(B) then F() is still a Dirichlet process. If =X is a weak Dirichlet process with finite quadratic variation, and F: C⁰,¹([0,T]× B) is of class C⁰,¹, then [F(t, ₜ)] is a weak Dirichlet process. We specify corresponding results when B=C([-τ,0]) and =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.