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July 2013 Central limit theorem for a Stratonovich integral with Malliavin calculus
Daniel Harnett, David Nualart
Ann. Probab. 41(4): 2820-2879 (July 2013). DOI: 10.1214/12-AOP769

Abstract

The purpose of this paper is to establish the convergence in law of the sequence of “midpoint” Riemann sums for a stochastic process of the form $f'(W)$, where $W$ is a Gaussian process whose covariance function satisfies some technical conditions. As a consequence we derive a change-of-variable formula in law with a second order correction term which is an Itô integral of $f"(W)$ with respect to a Gaussian martingale independent of $W$. The proof of the convergence in law is based on the techniques of Malliavin calculus and uses a central limit theorem for $q$-fold Skorohod integrals, which is a multi-dimensional extension of a result proved by Nourdin and Nualart [J. Theoret. Probab. 23 (2010) 39–64]. The results proved in this paper are generalizations of previous work by Swanson [Ann. Probab. 35 (2007) 2122–2159] and Nourdin and Réveillac [Ann. Probab. 37 (2009) 2200–2230], who found a similar formula for two particular types of bifractional Brownian motion. We provide three examples of Gaussian processes $W$ that meet the necessary covariance bounds. The first one is the bifractional Brownian motion with parameters $H\le1/2$, $HK=1/4$. The others are Gaussian processes recently studied by Swanson [Probab. Theory Related Fields 138 (2007) 269–304], [Ann. Probab. 35 (2007) 2122–2159] in connection with the fluctuation of empirical quantiles of independent Brownian motion. In the first example the Gaussian martingale is a Brownian motion, and expressions are given for the other examples.

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Daniel Harnett. David Nualart. "Central limit theorem for a Stratonovich integral with Malliavin calculus." Ann. Probab. 41 (4) 2820 - 2879, July 2013. https://doi.org/10.1214/12-AOP769

Information

Published: July 2013
First available in Project Euclid: 3 July 2013

zbMATH: 1285.60050
MathSciNet: MR3112933
Digital Object Identifier: 10.1214/12-AOP769

Subjects:
Primary: 60F05, 60G15, 60H05, 60H07

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 4 • July 2013
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