The multivariate fractional Ornstein-Uhlenbeck process

TL;DR

This paper introduces a multivariate fractional Ornstein-Uhlenbeck process with different Hurst exponents, characterizes its properties, and develops inference methods with asymptotic analysis using advanced probabilistic techniques.

Contribution

It defines a new multivariate Gaussian process with flexible dependence and Hurst parameters, and proposes novel estimators with proven asymptotic properties.

Findings

The process is stationary and ergodic with explicit correlation structure.

Two estimators are proposed and their asymptotic behaviors are characterized.

Numerical experiments support the theoretical results and suggest further applications.

Abstract

Starting from the notion of multivariate fractional Brownian Motion introduced in [F. Lavancier, A. Philippe, and D. Surgailis. Covariance function of vector self-similar processes. Statistics & Probability Letters, 2009] we define a multivariate version of the fractional Ornstein-Uhlenbeck process. This multivariate Gaussian process is stationary, ergodic and allows for different Hurst exponents on each component. We characterize its correlation matrix and its short and long time asymptotics. Besides the marginal parameters, the cross correlation between one-dimensional marginal components is ruled by two parameters. We consider the problem of their inference, proposing two types of estimator, constructed from discrete observations of the process. We establish their asymptotic theory, in one case in the long time asymptotic setting, in the other case in the infill and long time asymptotic setting. The limit behavior can be asymptotically Gaussian or non-Gaussian, depending on the values of the Hurst exponents of the marginal components. The technical core of the paper relies on the analysis of asymptotic properties of functionals of Gaussian processes, that we establish using Malliavin calculus and Stein's method. We provide numerical experiments that support our theoretical analysis and also suggest a conjecture on the application of one of these estimators to the multivariate fractional Brownian Motion.