Continuous-time autoregressive moving-average processes in Hilbert space

TL;DR

This paper introduces and analyzes continuous-time autoregressive moving-average (CARMA) processes in Hilbert spaces driven by Levy processes, establishing their properties and connections to stochastic wave equations.

Contribution

It extends CARMA processes to infinite-dimensional Hilbert spaces and links them to stochastic wave equations and functional autoregressive models.

Findings

Defined CARMA processes in Hilbert spaces

Established properties and equivalences of these processes

Connected CARMA processes to stochastic wave equations

Abstract

We introduce the class of continuous-time autoregressive moving-average (CARMA) processes in Hilbert spaces. As driving noises of these processes we consider Levy 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.