A central limit theorem for the realised covariation of a bivariate Brownian semistationary process

TL;DR

This paper establishes a central limit theorem for the scaled realised covariation of a bivariate Brownian semistationary process, extending asymptotic theory beyond classical semimartingale models using Malliavin calculus.

Contribution

It introduces a novel asymptotic framework for bivariate Brownian semistationary processes outside the semimartingale setting, utilizing advanced Malliavin calculus techniques.

Findings

Proves a weak law of large numbers for the process

Derives a central limit theorem in a multivariate context

Extends theory beyond classical semimartingale assumptions

Abstract

This article presents a weak law of large numbers and a central limit theorem for the scaled realised covariation of a bivariate Brownian semistationary process. The novelty of our results lies in the fact that we derive the suitable asymptotic theory both in a multivariate setting and outside the classical semimartingale framework. The proofs rely heavily on recent developments in Malliavin calculus.