Necessity of weak subordination for some strongly subordinated L\'evy processes

TL;DR

This paper investigates conditions under which strong and weak subordination of multivariate Lévy processes are equivalent, establishing when strong subordination yields a Lévy process and the necessity of weak subordination in certain cases.

Contribution

It proves that strong and weak subordination are equal in law under specific conditions and identifies cases where strong subordination must coincide with weak subordination.

Findings

Strong and weak subordination are equal in law under certain conditions.

Strong subordination produces a Lévy process when the subordinator is deterministic or pure-jump with finite activity.

Weak subordination extends strong subordination to cases where it may not produce a Lévy process.

Abstract

Consider the strong subordination of a multivariate L\'evy process with a multivariate subordinator. If the subordinate is a stack of independent L\'evy processes and the components of the subordinator are indistinguishable within each stack, then strong subordination produces a L\'evy process, otherwise it may not. Weak subordination was introduced to extend strong subordination, always producing a L\'evy process even when strong subordination does not. Here, we prove that strong and weak subordination are equal in law under the aforementioned condition. In addition, we prove that if strong subordination is a L\'evy process, then it is necessarily equal in law to weak subordination in two cases: firstly, when the subordinator is deterministic and secondly, when it is pure-jump with finite activity.