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Processus de Markov déterministes par morceaux branchants et problème d’arrêt optimal, application à la division cellulaire

Abstract : Piecewise deterministic Markov processes (PDMP) form a large class of stochastic processes characterized by a deterministic evolution between random jumps. They fall into the class of hybrid processes with a discrete mode and an Euclidean component (called the state variable). Between the jumps, the continuous component evolves deterministically, then a jump occurs and a Markov kernel selects the new value of the discrete and continuous components. In this thesis, we extend the construction of PDMPs to state variables taking values in some measure spaces with infinite dimension. The aim is to model cells populations keeping track of the information about each cell. We study our measured-valued PDMP and we show their Markov property. With thoses processes, we study a optimal stopping problem. The goal of an optimal stopping problem is to find the best admissible stopping time in order to optimize some function of our process. We show that the value fonction can be recursively constructed using dynamic programming equations. We construct some epsilon-optimal stopping times for our optimal stopping problem. Then, we study a simple finite-dimension real-valued PDMP, the TCP process. We use Euler scheme to approximate it, and we estimate some types of errors. We illustrate the results with numerical simulations.
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Submitted on : Monday, November 25, 2019 - 10:55:09 AM
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Maud Joubaud. Processus de Markov déterministes par morceaux branchants et problème d’arrêt optimal, application à la division cellulaire. Statistiques [math.ST]. Université Montpellier, 2019. Français. ⟨NNT : 2019MONTS031⟩. ⟨tel-02378390⟩

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