A Multiplicative Ergodic Theorem for Discontinuous Random Dynamical Systems and Applications

TL;DR

This paper establishes a multiplicative ergodic theorem for linear cocycles with jump discontinuities driven by non-Gaussian Lévy noise, extending spectral theory to more general stochastic systems with applications to Lévy-driven models.

Contribution

It introduces a new ergodic theorem applicable to discontinuous linear cocycles influenced by Lévy noise, broadening the scope of spectral analysis in stochastic systems.

Findings

Proved a multiplicative ergodic theorem for jump discontinuous cocycles.

Applied the theorem to systems with Lévy motions.

Extended spectral properties to non-Gaussian noise-driven systems.

Abstract

Motivated by studying stochastic systems with non-Gaussian L\'evy noise, spectral properties for a type of linear cocycles are considered. These linear cocycles have countable jump discontinuities in time. A multiplicative ergodic theorem is proved for such linear cocycles. Then, the result is illustrated for two linear stochastic systems with general L\'evy motions.