Law equivalence of Ornstein--Uhlenbeck processes driven by a L\'evy process

TL;DR

This paper establishes conditions under which Ornstein-Uhlenbeck processes driven by Lévy processes have equivalent laws, and shows that for purely jump Lévy processes, absolute continuity implies almost sure equality of solutions.

Contribution

It provides new criteria for law equivalence and absolute continuity of Ornstein-Uhlenbeck processes driven by Lévy processes, extending understanding of their probabilistic behavior.

Findings

Law equivalence holds when covariance eigenvalues are positive.

Absolute continuity implies almost sure equality for purely jump Lévy-driven processes.

Conditions depend on the structure of the Lévy process and covariance operator.

Abstract

We demonstrate that two Ornstein--Uhlenbeck processes, that is, solutions to certain stochastic differential equations that are driven by a L\'evy process L have equivalent laws as long as the eigenvalues of the covariance operator associated to the Wiener part of L are strictly positive. Moreover, we show that in the case where the underlying L\'evy process is a purely jump process, which means that neither it has a Wiener part nor the drift, the absolute continuity of the law of one solution with respect to another forces equality of the solutions almost surely.