Mathematics – Probability
Scientific paper
2011-05-24
Mathematics
Probability
41 pages
Scientific paper
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 of 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 R\'evaillac [7], who found a similar formula for two particular types of fractional 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 variance equal to t^2.
Harnett Daniel
Nualart David
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