Continuous-Time Autoregressive Moving-Average Processes in Hilbert Space

Abstract

We introduce the class of continuous-time autoregressive moving-average (CARMA) processes in Hilbert spaces. As driving noises of these processes we consider Lévy processes in Hilbert space. We provide the basic definitions, show relevant properties of these processes and establish the equivalents of CARMA processes on the real line. Finally, CARMA processes in Hilbert space are linked to the stochastic wave equation and functional autoregressive processes.

On non-negative modeling with CARMA processes

“…Proposition 4.1). Many results about CARMA processes are given in the literature, for example, a prediction formula, a multivariate extension, noise recovery, an extension to Hilbert space valued processes, an extension to incorporate long memory, and a CAR(∞) representation (see [4,9,14,15,17,30]). Moreover, there exists a large body of literature on the application of CARMA processes.…”