Comparison of time-inhomogeneous Markov processes

TL;DR

This paper develops comparison results for time-inhomogeneous Markov processes using generator conditions and invariance properties, extending existing theory and applying to diffusions, processes with independent increments, and Lévy-driven diffusions.

Contribution

It introduces a novel representation for solutions of inhomogeneous evolution problems, enabling new comparison results for a broad class of Markov processes.

Findings

Comparison results for diffusions and Lévy processes.

Extension of generator-based comparison to unbounded function classes.

Application to various time-inhomogeneous stochastic processes.

Abstract

Comparison results are given for time-inhomogeneous Markov processes with respect to function classes induced stochastic orderings. The main result states comparison of two processes, provided that the comparability of their infinitesimal generators as well as an invariance property of one process is assumed. The corresponding proof is based on a representation result for the solutions of inhomogeneous evolution problems in Banach spaces, which extends previously known results from the literature. Based on this representation, an ordering result for Markov processes induced by bounded and unbounded function classes is established. We give various applications to time-inhomogeneous diffusions, to processes with independent increments and to L\'evy driven diffusion processes.