Weak Approximations for Wiener Functionals

Abstract

In this paper we introduce a simple space-filtration discretization scheme on Wiener space which allows us to study weak decompositions and smooth approximations for a large class of Wiener functionals. We show that any Wiener functional has an underlying robust semimartingale skeleton which under mild conditions converges to it. The approximation is given in terms of discrete-jumping filtrations which allow us to approximate irregular processes by means of a stochastic derivative operator on Wiener space. As a by-product, we prove that continuous paths and a suitable notion of energy are sufficient in order to get a unique orthogonal decomposition similar to weak Dirichlet processes. In this direction, we generalize the main results given in Graversen and Rao [29] and Coquet et al. [12] in the particular Brownian filtration case. The second part of this paper is devoted to the application of these abstract results to concrete non-smooth processes. We show that our embedded semimartingale structure provides an easily implementable approximation scheme for the classical Clark-Ocone formula in full generality. Unlike in previous works our methodology does not assume an underlying Markovian structure and requires no use of Malliavin weights as in the classical literature of Mathematical Finance.