  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.
