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

TL;DR

This paper establishes conditions under which the Stratonovich integral of a smooth function of a Gaussian process converges weakly, extending known results for fractional Brownian motion to a broader class of Gaussian processes.

Contribution

It provides new covariance conditions ensuring weak convergence of the Stratonovich integral for a wider class of Gaussian processes, along with a change-of-variable formula in law.

Findings

Convergence in law of the Stratonovich integral under specified covariance conditions

Extension of known results from fractional Brownian motion to other Gaussian processes

Use of Malliavin calculus and a central limit theorem in the proof

Abstract

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.