We propose a numerical method to approximate the value function for the optimal stopping problem of a piecewise deterministic Markov process (PDMP). Our approach is based on quantization of the post jump location-inter-arrival time Markov chain naturally embedded in the PDMP, and path-adapted time discretization grids. It allows us to derive bounds for the convergence rate of the algorithm and to provide a computable ǫ-optimal stopping time. The paper is illustrated by a numerical example.
If the reliability community remains interested in dynamic reliability theory, it is not really convinced by the ability of already available approaches to treating current problems from within the operational domain, even if the methodological quality of these approaches is undeniable. This paper is in keeping with two papers presented in earlier conferences. Its aim is to show the potentialities of a method that combines the high modelling capacity of the piecewise-deterministic processes with the great computing power inherent in the Monte Carlo simulation. This method has been applied to a well-known test-case example to test its ability to solve common dynamic reliability problems. Two sets of results have been obtained. The first one has been compared to those coming from a Petri-net model to obtain a preliminary validation of the proposed method. The second one, related to a more complex case, has been compared to already published results found in the literature. Contrary to already existing methods, the approach here is an exact Monte Carlo sampling method; it does not need timespace discretization.
The aim of this chapter is to present the computational method developed in [DES 10] for the control of a Piecewise-Deterministic Markov Process (PDMP) (X t) t 0 and to explain how similar ideas can be used for statistical inference. Most technical details are omitted to focus on the practical application of the procedure: we state in details the class of PDMPs it can be applied to and give all the algorithms necessary to its implementation. The main material of this chapter was originally published as [DES 10] and [BRA 12b]. Additional details and examples can also be found in [DES 15].
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