Central limit theorem for a Stratonovich integral with Malliavin calculus

TL;DR

This paper proves a central limit theorem for midpoint Riemann sums of a Gaussian process, leading to a change-of-variable formula with a second order correction, using Malliavin calculus techniques.

Contribution

It generalizes previous results by establishing a CLT for a broader class of Gaussian processes and deriving a law in law change-of-variable formula with a second order correction term.

Findings

Convergence in law of midpoint Riemann sums for Gaussian processes.

Derivation of a change-of-variable formula with a second order correction.

Application to bifractional Brownian motion and other Gaussian processes.

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 It\^{o} 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 R\'{e}veillac [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.