Weak approximations for Wiener functionals

TL;DR

This paper introduces a simple discretization scheme on Wiener space for approximating Wiener functionals, enabling robust weak decompositions, explicit approximations, and numerical schemes without relying on Markovian assumptions or Malliavin calculus.

Contribution

It presents a novel discretization approach that approximates Wiener functionals and weak Dirichlet processes, applicable to non-Markovian systems, with explicit schemes and convergence guarantees.

Findings

Provides a robust semimartingale skeleton for Wiener functionals.

Develops an implementable approximation for the Clark-Ocone formula.

Proposes a method to compute optimal stopping times in non-Markovian contexts.

Abstract

In this paper we introduce a simple space-filtration discretization scheme on Wiener space which allows us to study weak decompositions and smooth explicit 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 discretization is given in terms of discrete-jumping filtrations which allow us to approximate nonsmooth processes by means of a stochastic derivative operator on the Wiener space. As a by-product, we provide a robust semimartingale approximation for weak Dirichlet-type processes. The underlying semimartingale skeleton is intrinsically constructed in such way that all the relevant structure is amenable to a robust numerical scheme. In order to illustrate the results, we provide 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 does not require Malliavin weights. We conclude by proposing a method that enables us to compute optimal stopping times for possibly non-Markovian systems arising, for example, from the fractional Brownian motion.