Optimal stopping for measure-valued piecewise deterministic Markov processes

TL;DR

This paper studies optimal stopping problems for measure-valued PDMPs, motivated by population monitoring, and develops a dynamic programming approach to compute the value function, highlighting differences between controlling entire populations versus lineages.

Contribution

It introduces a framework for optimal stopping of measure-valued PDMPs and establishes a dynamic programming principle for these processes.

Findings

Value function obtained via iterative dynamic programming.

Controlling the whole population differs from controlling a lineage.

Counter-example illustrating the distinction in control strategies.

Abstract

This paper investigates the random horizon optimal stopping problem for measure-valued piecewise deterministic Markov processes (PDMPs). This is motivated by population dynamics applications, when one wants to monitor some characteristics of the individuals in a small population. The population and its individual characteristics can be represented by a point measure. We first define a PDMP on a space of locally finite measures. Then we define a sequence of random horizon optimal stopping problems for such processes. We prove that the value function of the problems can be obtained by iterating some dynamic programming operator. Finally we prove on a simple counter-example that controlling the whole population is not equivalent to controlling a random lineage.