Measure-Valued Generators of General Piecewise Deterministic Markov Processes

TL;DR

This paper develops a measure-valued generator theory for general PDMPs with complex inter-occurrence times, providing a new analytical framework and applications to expected discounted values.

Contribution

It introduces the measure-valued generator for general PDMPs, extending the domain to locally finite variation functions and linking it to additive functionals and integro-differential equations.

Findings

Characterization of additive functionals via measure-valued generators

Conditions for local martingale and semimartingale properties

Derivation of measure integro-differential equations for expected values

Abstract

We consider a piecewise-deterministic Markov process (PDMP) with general conditional distribution of inter-occurrence time, which is called a general PDMP here. Our purpose is to establish the theory of measure-valued generator for general PDMPs. The additive functional of a semi-dynamic system (SDS) is introduced firstly, which presents us an analytic tool for the whole paper. The additive functionals of a general PDMP are represented in terms of additive functionals of the SDS. The necessary and sufficient conditions of being a local martingale or a special semimartingale for them are given. The measure-valued generator for a general PDMP is introduced, which takes value in the space of additive functionals of the SDS. And its domain is completely described by analytic conditions. The domain is extended to the locally (path-)finite variation functions. As an application of measure-valued generator, we study the expected cumulative discounted value of an additive functional of the general PDMP, and get a measure integro-differential equation satisfied by the expected cumulative discounted value function.