Measure-Valued CARMA Processes

TL;DR

This paper introduces measure-valued CARMA processes driven by Lévy subordinators, establishing their existence, properties, and stationarity conditions, with applications to modeling spatio-temporal random fields.

Contribution

It defines measure-valued CARMA processes as solutions to linear state-space models in Banach spaces, extending CARMA theory to measure-valued processes with broad applicability.

Findings

Proves existence and cone-invariance of measure-valued CARMA processes

Derives explicit stationarity conditions for these processes

Shows their applicability to modeling spatio-temporal random fields

Abstract

In this paper, we examine continuous-time autoregressive moving-average (CARMA) processes on Banach spaces driven by L\'evy subordinators. We show their existence and cone-invariance, investigate their first and second order moment structure, and derive explicit conditions for their stationarity. Specifically, we define a measure-valued CARMA process as the analytically weak solution of a linear state-space model in the Banach space of finite signed measures. By selecting suitable input, transition, and output operators in the linear state-space model, we show that the resulting solution possesses CARMA dynamics and remains in the cone of positive measures defined on some spatial domain. We also illustrate how positive measure-valued CARMA processes can be used to model the dynamics of functionals of spatio-temporal random fields and connect our framework to existing CARMA-type models from the literature, highlighting its flexibility and broader applicability.