Modeling stationary data by a class of generalised Ornstein-Uhlenbeck processes

TL;DR

This paper introduces a new class of higher order Ornstein-Uhlenbeck processes, OU(p), which better model stationary data than traditional AR(p) models, especially when data is sampled at regular intervals.

Contribution

The paper develops a novel framework for iterating OU processes to create OU(p) models, providing explicit covariance formulas and efficient parameter estimation methods.

Findings

OU(p) models outperform AR(p) models on real data.

Explicit covariance formulas facilitate parameter estimation.

Iterated OU processes can be expressed as linear combinations of basic OU processes.

Abstract

An Ornstein-Uhlenbeck (OU) process can be considered as a continuous time interpolation of the discrete time AR$(1)$ process. Departing from this fact, we analyse in this work the effect of iterating OU treated as a linear operator that maps a Wiener process onto Ornstein-Uhlenbeck process, so as to build a family of higher order Ornstein-Uhlenbeck processes, OU$(p)$, in a similar spirit as the higher order autoregressive processes AR$(p)$. We show that for $p \ge 2$ we obtain in general a process with covariances different than those of an AR$(p)$, and that for various continuous time processes, sampled from real data at equally spaced time instants, the OU$(p)$ model outperforms the appropriate AR$(p)$ model. Technically our composition of the OU operator is easy to manipulate and its parameters can be computed efficiently because, as we show, the iteration of OU operators leads to a process that can be expressed as a linear combination of basic OU processes. Using this expression we obtain a closed formula for the covariance of the iterated OU process, and consequently estimate the parameters of an OU$(p)$ process by maximum likelihood or, as an alternative, by matching correlations, the latter being a procedure resembling the method of moments.