Comparison of Markov processes via infinitesimal generators

Abstract

We derive comparison results for Markov processes with respect to stochastic orderings induced by function classes. Our main result states that stochastic monotonicity of one process and comparability of the infinitesimal generators implies ordering of the processes. Unlike in previous work no boundedness assumptions on the function classes are needed anymore. We also present an integral version of the comparison result which does not need the local comparability assumption of the generators. The method of proof is also used to derive comparison results for time-discrete Markov processes.

A note on applications of stochastic ordering to control problems in insurance and finance

“…the increasing convex ordering by dominating one process by convex combination of other processes. For a recent paper on the stochastic comparison of Markov processes, see [8]. However, note that our processes are not necessarily Markovian and allow a comparison only after a time change.…”

Comparison of time-inhomogeneous Markov processes

“…Note that the autonomous case of the weak evolution problem is a weakening of the inhomogeneous Cauchy problem. The basic representation result for solutions of the inhomogeneous Cauchy equation in a Banach space was used in Rüschendorf and Wolf (2011) to establish comparison results for homogeneous Markov processes. The following is an extension of this representation result to the weak evolution problem.…”

“…Rüschendorf (2008) established a comparison result for homogeneous Markov processes using boundedness conditions on the order defining function classes. Comparison results for homogeneous Markov processes with transition functions defined on general Banach spaces are given in Rüschendorf and Wolf (2011). The results are based on an integral representation of solutions to the inhomogeneous Cauchy problem.…”

Comparison of Markov processes by the martingale comparison method

“…Stochastic ordering and comparison results for Markov processes are a basic problem of probability theory. They have a long history and are motivated by a number of applications in a variety of fields (see Massey (1987), Cox et al (1996), Daduna and Szekli (2006), Rüschendorf (2008), Krasin and Melnikov (2009), Rüschendorf and Wolf (2011), Rüschendorf et al (2016), Criens (2017) and Criens (2019)). Various approaches ranging from analytic to coupling methods have been developed to this aim sometimes in the context of specific models or specific applications.…”