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 $\mathcal{O}\mathcal{U}_{\kappa }$ that maps a random variable y(t) onto $\mathcal{O}\mathcal{U}_{\kappa }y(t) =\int _{ -\infty }^{t}\mathrm{e}^{-\kappa (t-s)}dy(s)$. 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.