CARMA(p,q) generalized random processes

There is now a vast literature on the theory and applications of generalized random processes, pioneered by Itô (1953), Gel'fand (1955) and Yaglom (1957). In this note we make use of the theory of generalized random processes as defined in the book of Gel'fand and Vilenkin (1964) to extend the definition of continuous-time ARMA(p,q) processes to allow q≥p, in which case the processes do not exist in the classical sense. The resulting CARMA generalized random processes provide a framework within which it is possible to study derivatives of CARMA processes of arbitrarily high order.