The stochastic Cauchy problem driven by a cylindrical Levy process

TL;DR

This paper establishes conditions for solutions to stochastic Cauchy problems driven by cylindrical Levy processes, introducing a stochastic Fubini theorem and analyzing solution properties like Markov and continuity.

Contribution

It provides necessary and sufficient conditions for solutions and develops a stochastic Fubini theorem for cylindrical Levy processes, advancing the understanding of such stochastic equations.

Findings

Solutions have almost surely scalarly square integrable paths

The solution process exhibits the Markov property

Solutions are stochastically continuous

Abstract

In this work, we derive sufficient and necessary conditions for the existence of a weak and mild solution of an abstract stochastic Cauchy problem driven by an arbitrary cylindrical Levy process. Our approach requires to establish a stochastic Fubini result for stochastic integrals with respect to cylindrical Levy processes. This approach enables us to conclude that the solution process has almost surely scalarly square integrable paths. Further properties of the solution such as the Markov property and stochastic continuity are derived.