Levy-driven CARMA Random Fields on Rn

We define an isotropic Levy-driven CARMA(p,q) random field on Rn as the integral of an isotropic CARMA kernel with respect to a Levy sheet. Such fields constitute a parametric family characterized by an autoregressive polynomial a and a moving average polynomial b having zeros in both the left and right complex half-planes. They extend the well-balanced Ornstein-Uhlenbeck process of Schnurr and Woerner (2011) to a well-balanced CARMA process in one dimension (with a much richer class of autocovariance functions) and to an isotropic CARMA random field on Rn for n > 1. We derive second-order properties of these random fields and find that CAR(1) constitutes a subclass of the well known Matern class. If the driving Levy sheet is compound Poisson it is a trivial matter to simulate the corresponding random field on any n-dimensional hypercube. Joint estimation of CARMA kernel parameters and knots locations is proposed in cases driven by compound Poisson sheets and is illustrated by applications to land price data in Tokyo as well as simulated data.