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 
, 
provided that the integral makes sense.
In particular, when 
is replaced by a Levy process 
on 
, 
, 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 
. We denote by 
the families of processes obtained by applying successively 
OU operators 
, 
to . 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 
obtained from 
has the same second order moments of an ARMA
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.