Continuous-time locally stationary time series models

TL;DR

This paper extends the concept of locally stationary processes to continuous-time, introducing a spectral density framework and analyzing time-varying L\'evy-driven processes, including CARMA models.

Contribution

It develops a continuous-time locally stationary process framework, deriving a spectral density and analyzing time-varying L\'evy-driven state space and CARMA processes.

Findings

Established a continuous-time locally stationary process definition.

Derived a unique time-varying spectral density using Wigner-Ville spectrum.

Provided conditions for local stationarity of time-varying L\'evy-driven processes.

Abstract

We adapt the classical definition of locally stationary processes in discrete-time to the continuous-time setting and obtain equivalent representations in the time and frequency domain. From this, a unique time-varying spectral density is derived using the Wigner-Ville spectrum. As an example, we investigate time-varying L\'evy-driven state space processes, including the class of time-varying L\'evy-driven CARMA processes. First, the connection between these two classes of processes is examined. Considering a sequence of time-varying L\'evy-driven state space processes, we then give sufficient conditions on the coefficient functions that ensure local stationarity with respect to the given definition.