Weak convergence of the Stratonovich integral with respect to a class of Gaussian processes

Details Weak convergence of the Stratonovich integral with respect to a class of Gaussian processes Weak convergence of the Stratonovich integral with respect to a class of Gaussian processes

2011-09-14

arxiv.org/abs/1109.3110v1

Mathematics

Probability

39 pages

Scientific paper

For a Gaussian process X and smooth function f, we define the Stratonovich integral of f with respect to X as the limit, if it exists, of a sequence of trapezoidal Riemann sums. We present a set of covariance conditions on X under which the sequence of Riemann sums converges in law. As a consequence, we derive a change-of-variable formula in law with a correction term which is an It\^o integral of f''' with respect to a Gaussian martingale independent of X. The proof uses Malliavin calculus and follows from a central limit theorem first proved by Nourdin and Nualart in [9]. The results generalize recent work by Nourdin, R\'eveillac and Swanson [10], who proved this formula for fractional Brownian motion with H=1/6. In this paper, we extend to a larger class of Gaussian processes, which includes bifractional Brownian motion with HK=1/6 and subfractional Brownian motion with h=1/3.

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