# Modeling stationary data by classes of generalized Ornstein-Uhlenbeck processes. (A. Cabaña)

###### Modeling stationary data by classes of generalized Ornstein-Uhlenbeck processes, A. Cabaña (presenter), E. M. Cabaña and A. Arratia

A work presented at the 7èmes Journées Statistiques du Sud, Barcelona, June 9-11, 2014

Abstract: Consider the complex Ornstein Uhlenbeck operator defined by $OU_{\kappa}y(t)=\int_{-\infty}^t{\rm e}^{-\kappa(ts)}$, $\kappa\in{\rm\bf C},\Re(\kappa)>0$ provided that the integral makes sense.
In particular, when $y$ is replaced by a Levy process $\Lambda$ on ${R}$, $x(t)=OU_{\kappa}\Lambda(t)$, is Levy-driven Ornstein Uhlembeck process. New stationary processes can be constructed by iterating the application of OU to Levy processes, in particular, to the Wiener process $w$. We denote by $OU(p)$ the families of processes obtained by applying successively $p$ OU operators $OU_{\kappa_j}$, $j=1,2,\dots,p$ to $\Lambda$. These processes can be used as models for stationary continuous parameter processes, and their discretized version, for stationary time series.
We show in particular that the series $x(0), x(1), \dots, x(n)$ obtained from $x=\prod_{j=1}^p{\cal OU}_{\kappa_j}(\Lambda)$ has the same second order moments of an ARMA$(p,p-1)$ model hence, it is an ARMA when the driving process is a Wiener process.
We present an empirical comparison of the abilities of OU and ARMA
models to fit stationary data. In particular, we show examples of processes with long dependence where the fitting of the empirical autocorrelations is improved by using OU process rather than ARMA models, and with fewer parameters in the former case.