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

For a Gaussian process X and smooth function f , we consider a Stratonovich integral of f ( X ) , defined as the weak limit, if it exists, of a sequence of Riemann sums. We give covariance conditions on X such that the sequence converges in law. This gives a change-of-variable formula in law with a correction term which is an Itô integral of f ‴ with respect to a Gaussian martingale independent of X . The proof uses Malliavin calculus and a central limit theorem from Nourdin and Nualart (2010) [8]. This formula was known for fBm with H = 1 / 6 Nourdin et al. (2010) [9]. We extend this to a larger class of Gaussian processes.