Spectral Representation of Multivariate Regularly Varying L\'evy and CARMA processes

TL;DR

This paper develops a spectral framework for multivariate regularly varying Lévy and CARMA processes, providing new insights into their properties, especially in the infinite variance case, and extending existing models.

Contribution

It introduces a spectral representation for regularly varying Lévy processes with index between one and two, enabling a spectral definition of multivariate Lévy-driven CARMA processes.

Findings

Spectral representation for Lévy processes with index between one and two.

Extension of CARMA processes to infinite variance cases.

Connection to existing models with finite second moments.

Abstract

A spectral representation for regularly varying L\'evy processes with index between one and two is established and the properties of the resulting random noise are discussed in detail giving also new insight in the $L^2$-case where the noise is a random orthogonal measure. This allows a spectral definition of multivariate regularly varying L\'evy-driven continuous time autoregressive moving average (CARMA) processes. It is shown that they extend the well-studied case with finite second moments and coincide with definitions previously used in the infinite variance case when they apply.