Stochastic evolution equations driven by arbitrary cylindrical L\'evy processes

TL;DR

This paper proves existence and uniqueness of mild solutions for stochastic evolution equations driven by cylindrical Le9vy processes in Hilbert spaces without moment assumptions, using a novel approximation scheme.

Contribution

It introduces a pathwise Euler-Peano approximation method to handle cylindrical Le9vy noise without classical stochastic calculus tools.

Findings

Established the first existence and uniqueness results for such equations.

Developed a noise-dependent stopping time approximation scheme.

Provided a framework for multiplicative cylindrical Le9vy noise in infinite dimensions.

Abstract

We establish the first existence and uniqueness result for mild solutions of abstract stochastic evolution equations driven by arbitrary cylindrical L\'evy processes in Hilbert spaces. The coefficients are assumed to satisfy global Lipschitz conditions, and no moment assumptions are imposed on the driving noise. The principal difficulty arises from the fact that cylindrical L\'evy processes exist solely in a generalised sense and typically admit no semimartingale or L\'evy-It\^o decomposition, which precludes the use of classical existence methods. To overcome these obstacles, we develop a pathwise adaptive Euler-Peano approximation scheme based on noise-dependent stopping times and a fixed-point formulation of the mild solution operator. The resulting approach avoids stochastic calculus techniques relying on semimartingale decompositions and provides a robust and flexible framework for treating multiplicative cylindrical L\'evy noise in infinite-dimensional systems.