Stochastic integration with respect to cylindrical L\'evy processes

TL;DR

This paper develops a new stochastic integral for cylindrical Lévy processes in Hilbert spaces, enabling integration with more general integrands without moment or boundedness restrictions.

Contribution

It introduces a novel stochastic integration framework for cylindrical Lévy processes, expanding the scope beyond traditional semi-martingale approaches.

Findings

Integral process is an adapted Hilbert space semi-martingale

Integrands are adapted stochastic processes in Hilbert-Schmidt operators

No moment or boundedness conditions required for integrands or integrator

Abstract

A cylindrical Levy process does not enjoy a cylindrical version of the semi-martingale decomposition which results in the need to develop a completely novel approach to stochastic integration. In this work, we introduce a stochastic integral for random integrands with respect to cylindrical Levy processes in Hilbert spaces. The space of admissible integrands consists of adapted stochastic processes with values in the space of Hilbert-Schmidt operators. Neither the integrands nor the integrator is required to satisfy any moment or boundedness condition. The integral process is characterised as an adapted, Hilbert space valued semi-martingale with cadlag trajectories.