Central Limit Theorem for a Stratonovich Integral with Malliavin Calculus

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-ofvariable 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 multidimensional extension of a result proved by Nourdin and Nualart in [5]. The results proved in this paper are generalizations of previous work by Swanson [10] and Nourdin and Réveillac [7], who found a similar formula for two particular types of bifractional Brownian motion. We provide two examples of Gaussian processes W that meet the necessary covariance bounds. The first one is the bifractional Brownian motion with parameters H ≤ 1/2, HK = 1/4. The second one is a Gaussian process recently studied by Swanson [9] in connection with the fluctuation of empirical quantiles of independent Brownian motion. In the first example the Gaussian martingale is a Brownian motion and in the second case it has a variance equal to t.