Generalized covariation and extended Fukushima decomposition for banach space-valued processes: Applications to windows of dirichlet processes

This paper is concerned with the notion of covariation for Banach space-valued processes. In particular, we introduce a notion of quadratic variation, which is a generalization of the classical restrictive formulation of Métivier and Pellaumail. Our approach is based on the notion of χ-covariation for processes with values in two Banach spaces B ₁ and B ₂, where χ is a suitable subspace of the dual of the projective tensor product of B ₁ and B ₂. We investigate some C ¹ type transformations for various classes of stochastic processes admitting a χ-quadratic variation and related properties. If ¹ and ² admit a χ-covariation, F ⁱ : B _i → , i = 1, 2 are of class C ¹ with some supplementary assumptions, then the covariation of the real processes F ¹( ¹) and F ²( ²) exist. A detailed analysis is provided on the so-called window processes. Let X be a real continuous process; the C([-τ, 0])-valued process X(·) defined by X _t(y) = X _t+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. Those will constitute a significant Fukushima decomposition for functionals of windows of (weak) Dirichlet processes. As application, we provide a new technique for representing a path-dependent random variable as its expectation plus a stochastic integral with respect to the underlying process.