Section: New Results

Optimal stopping for piecewise-deterministic Markov processes and applications

Participants : Adrien Brandejsky, Benoîte de Saporta, François Dufour, Huilong Zhang.

We worked further on numerical methods for optimal stopping of PDMPs. On the one hand, we applied our numerical method to compute an optimal maintenance date to the test case of the heated hold-up tank. The system consists of a tank containing a fluid whose level is controlled by three components: two inlet pumps and one outlet valve. A thermal power source heats up the fluid. The failure rates of the components depends on the temperature, the position of the three components monitors the liquid level in the tank and the liquid level determines the temperature. Therefore, this system can be modeled by a hybrid process where the discrete (components) and continuous (level, temperature) parts interact in a closed loop. We model the system by a piecewise deterministic Markov process, propose and implement a numerical method to compute the optimal maintenance date to repair the components before the total failure of the system. This work is published in [30] .

On the other hand, we investigated the optimal stopping problem under partial observations for PDMPs. We first obtain a recursive formulation of the optimal filter process and derive the dynamic programming equation of the partially observed optimal stopping problem. Then, we propose a numerical method, based on the quantization of the discrete-time filter process and the inter-jump times, to approximate the value function and to compute an $ϵ$-optimal stopping time. We prove the convergence of the algorithms and bound the rates of convergence. This work is published in [20] .