An alternative to CARMA models via iterations of Ornstein–Uhlenbeck processes

Abstract

We present a new construction of continuous ARMA processes based on iterating an Ornstein–Uhlenbeck operator OUκ that maps a random variable y(t) onto OUκ y(t) = ∫t–∞e–κ(t–s) dy(s). This construction resembles the procedure to build an AR(p) from an AR(1) and derives in a parsimonious model for continuous autoregression, with fewer parameters to compute than the known CARMA obtained as a solution of a system of stochastic differential equations. We show properties of this operator, give state space representation of the iterated Ornstein–Uhlenbeck process and show how to estimate the parameters of the model.