L\'evy-driven causal CARMA random fields

TL;DR

This paper introduces a new class of Le9vy-driven causal CARMA random fields on b4, extending traditional CARMA processes through stochastic partial differential equations, and explores their properties and sampling behavior.

Contribution

It generalizes CARMA processes to random fields on b4 using SPDEs, providing existence conditions and analyzing their second-order structure and sampling characteristics.

Findings

Existence of Le9vy-driven causal CARMA random fields established.

Conditions for the field to be an ARMA when sampled on a lattice derived.

Analysis of second-order structure and path properties provided.

Abstract

We introduce L\'evy-driven causal CARMA random fields on $\mathbb{R}^d$, extending the class of CARMA processes. The definition is based on a system of stochastic partial differential equations which generalize the classical state-space representation of CARMA processes. The resulting CARMA model differs fundamentally from the isotropic CARMA random field of Brockwell and Matsuda. We show existence of the model under mild assumptions and examine some of its features including the second-order structure and path properties. In particular, we investigate the sampling behavior and formulate conditions for the causal CARMA random field to be an ARMA random field when sampled on an equidistant lattice.