The Annals of Applied Probability

Weak approximations for Wiener functionals

Dorival Leão and Alberto Ohashi

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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.

Article information

Source
Ann. Appl. Probab., Volume 23, Number 4 (2013), 1660-1691.

Dates
First available in Project Euclid: 21 June 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1371834041

Digital Object Identifier
doi:10.1214/12-AAP883

Mathematical Reviews number (MathSciNet)
MR3098445

Zentralblatt MATH identifier
1284.60133

Subjects
Primary: 60Hxx: Stochastic analysis [See also 58J65]
Secondary: 60H20: Stochastic integral equations

Keywords
Weak convergence Clark–Ocone formula optimal stopping hedging

Citation

Leão, Dorival; Ohashi, Alberto. Weak approximations for Wiener functionals. Ann. Appl. Probab. 23 (2013), no. 4, 1660--1691. doi:10.1214/12-AAP883. https://projecteuclid.org/euclid.aoap/1371834041


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