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Optimal stopping for measure-valued piecewise deterministic Markov processes

Part of: Stochastic processes Markov processes Stochastic systems and control

Published online by Cambridge University Press:  16 July 2020

Bertrand Cloez ,
Benoîte de Saporta  and
Maud Joubaud
Bertrand Cloez*
Affiliation:
INRA
Benoîte de Saporta*
Affiliation:
University of Montpellier
Maud Joubaud*
Affiliation:
University of Montpellier
*
*Postal address: MISTEA, INRA, Montpellier SupAgro, University of Montpellier, Montpellier, France.
**Postal address: IMAG, Univ. Montpellier, CNRS, Montpellier, France. Email: benoite.de-saporta@umontpellier.fr
**Postal address: IMAG, Univ. Montpellier, CNRS, Montpellier, France. Email: benoite.de-saporta@umontpellier.fr
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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 via a simple counter-example that controlling the whole population is not equivalent to controlling a random lineage.

Keywords

Markov processesmeasure spaceoptimal stoppingdynamic programmingpopulation dynamics

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 92D25: Population dynamics (general) 93E20: Optimal stochastic control
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 2 , June 2020 , pp. 497 - 512
Copyright
© Applied Probability Trust 2020

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