Stochastic integration with respect to canonical $\alpha$-stable cylindrical L\'evy processes

TL;DR

This paper develops a new stochastic integration theory for symmetric alpha-stable cylindrical Lévy processes, overcoming the lack of semi-martingale structure by using decoupling inequalities and characterizing integrable processes.

Contribution

Introduces a novel integration framework for alpha-stable cylindrical Lévy processes using decoupling inequalities, characterizing integrable processes in the Bochner space.

Findings

Characterizes the largest space of integrable processes as predictable processes in L^α

Establishes a dominated convergence theorem for stochastic integrals

Demonstrates robustness of the integration theory in limit interchange

Abstract

In this work, we introduce a theory of stochastic integration with respect to symmetric $\alpha$-stable cylindrical L\'evy processes. Since $\alpha$-stable cylindrical L\'evy processes do not enjoy a semi-martingale decomposition, our approach is based on a decoupling inequality for the tangent sequence of the Radonified increments. This approach enables us to characterise the largest space of predictable Hilbert-Schmidt operator-valued processes which are integrable with respect to an $\alpha$-stable cylindrical L\'evy process as the collection of all predictable processes with paths in the Bochner space $L^\alpha$. We demonstrate the power and robustness of the developed theory by establishing a dominated convergence result allowing the interchange of the stochastic integral and limit.