Stochastic integration with respect to a L\'evy basis

TL;DR

This paper develops a comprehensive stochastic integration framework for predictable processes with respect to Lévy bases, extending deterministic measure theory to stochastic settings.

Contribution

It introduces a new integration theory based on tangent sequences, characterizes integrable processes via semimartingale characteristics, and proves continuity and convergence properties.

Findings

Characterizes integrable predictable processes using semimartingale characteristics.

Establishes a Musielak-Orlicz type structure for the space of integrands.

Proves continuity of the integral operator and a stochastic dominated convergence theorem.

Abstract

We develop a stochastic integration theory for predictable integrands with respect to a L\'evy basis. Our approach is based on decoupling inequalities for tangent sequences and reduces the construction of the stochastic integral essentially to the deterministic integration theory for infinitely divisible random measures developed by Rajput and Rosi\'nski. We characterise the corresponding class of integrable predictable processes in terms of the semimartingale characteristics associated with the driving random measure and show that the resulting space of integrands possesses a natural Musielak-Orlicz type structure equipped with an F-norm. Furthermore, we establish continuity properties of the integral operator and a stochastic version of Lebesgue's dominated convergence theorem.