The Annals of Probability

Central limit theorem for a Stratonovich integral with Malliavin calculus

Daniel Harnett and David Nualart

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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.

Article information

Ann. Probab., Volume 41, Number 4 (2013), 2820-2879.

First available in Project Euclid: 3 July 2013

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Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus 60F05: Central limit and other weak theorems 60G15: Gaussian processes

Itô formula Skorohod integral Malliavin calculus fractional Brownian motion


Harnett, Daniel; Nualart, David. Central limit theorem for a Stratonovich integral with Malliavin calculus. Ann. Probab. 41 (2013), no. 4, 2820--2879. doi:10.1214/12-AOP769.

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