Résumé
In our work, we study an optimal stopping problem for specific Markov processes: Piecewise Determinis-tic Markov Processes (PMDP). Those time-continuous processes were formalized by Davis in [2]. The only source of randomness in those Markov processes comes from the jumps. The jump times are drawn in a Poisson-like fashion. Between jumps, the process follows a deterministic trajectory, given by the flow of some differential equation. In Davis' work, the state space of PDMP is in R d for some integers d. We extend this vision to measure-valued processes. We introduce such measure-valued processes in order to model population dynamics problems, when the population size is small. The population characteristics can be described as follows: let n be the size of the population at some time t. Let x i be a real number, representing some biological trait of the i − th individual. It can be the size, the weight, the concentration of some protein... So, at this time t, information about the population can be summarized with a locally finite measure µ := n i=1 δ xi. The process, at time t, takes this value µ. It is related to a growth fragmentation model [1]. We investigate an optimal stopping problem for measure-valued PDMPs. The purpose is to select a stopping time τ in order to maximize some mean reward g of the PDMP (X t) t : sup τ E[g(X τ)]. To solve our optimal stopping problem, we imitate the technique from the paper [4]. This paper considers the specific case of R d-valued processes. The optimal performance is called the value function of the optimal stopping problem. We prove that this value function can be recursively constructed by iterating a dynamic programming operator. We illustrate this work with a toy model of cell division. In particular, we prove that controlling the whole population is not equivalent to controlling a tagged cell, unlike other classical problems [3]. Acknowledgement: The work was partially supported by Région Languedoc-Roussillon and FEDER under grant Chercheur(se)s d'Avenir, project PROMMECE.