Monotone and Convex Stochastic Orders for Processes with Independent Increments

TL;DR

This paper investigates monotone and convex stochastic orders for processes with independent increments, establishing relations between Poisson components and Lévy measures, and providing explicit conditions through coupling constructions.

Contribution

It introduces new relations between stochastic orders of Poisson components and Lévy measures, and offers explicit process characteristic conditions via couplings.

Findings

Relation between Poisson components and Lévy measures established

Explicit conditions on process characteristics derived

Coupling constructions used to prove stochastic order conditions

Abstract

We study monotone and convex stochastic orders for processes with independent increments. Our contributions are twofold: First, we relate stochastic orders of the Poisson component to orders of their (generalized) L\'evy measures. The relation is proven using an interpolation formula for infinitely divisible laws. Second, we derive explicit conditions on the characteristics of the processes. In this case, we prove the conditions via constructions of couplings.