Generalized covariation and extended Fukushima decompositions 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 Metivier and Pellaumail. Our approach is based on the notion of chi-covariation for processes with values in two Banach spaces B-1 and B-2, where chi 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.-quadratic variation and related properties. If X-1 and X-2 admit a chi-covariation, F-i : B-i -> 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 is provided on the so-called window processes. Let X be a real continuous process; the C([-tau, 0])-valued process X(.) defined by X-t(y) = Xt+y, where y is an element of [-tau, 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.