Central limit theorem for a Stratonovich integral with Malliavin calculus

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.