GENERALIZED COVARIATION AND EXTENDED FUKUSHIMA DECOMPOSITION FOR BANACH SPACE-VALUED PROCESSES: APPLICATIONS TO WINDOWS OF DIRICHLET PROCESSES
Abstract
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 B1 and B2, where χ is 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 𝕏1 and 𝕏2 admit a χ-covariation, Fi : Bi → ℝ, i = 1, 2 are of class C1 with some supplementary assumptions, then the covariation of the real processes F1(𝕏1) and F2(𝕏2) 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 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. 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.
