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 It\^o integral of $f"'$ with respect to a Gaussian martingale independent of $X$. The proof uses Malliavin calculus and a central limit theorem from [10]. This formula was known for fBm with $H=1/6$ [9]. We extend this to a larger class of Gaussian processes.