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)f′(W), where WW 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)f"(W) with respect to a Gaussian martingale independent of WW. The proof of the convergence in law is based on the techniques of Malliavin calculus and uses a central limit theorem for qq-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 WW that meet the necessary covariance bounds. The first one is the bifractional Brownian motion with parameters H≤1/2H≤1/2, HK=1/4HK=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.