scholarly journals Piecewise Deterministic Markov Processes for Continuous-Time Monte Carlo

2018 ◽  
Vol 33 (3) ◽  
pp. 386-412 ◽  
Author(s):  
Paul Fearnhead ◽  
Joris Bierkens ◽  
Murray Pollock ◽  
Gareth O. Roberts
Keyword(s):  
Monte Carlo ◽  
Markov Processes ◽  
Continuous Time ◽  
Piecewise Deterministic Markov Processes

scholarly journals Piecewise deterministic Markov processes for scalable Monte Carlo on restricted domains

2018 ◽  
Vol 136 ◽  
pp. 148-154 ◽  
Author(s):  
Joris Bierkens ◽  
Alexandre Bouchard-Côté ◽  
Arnaud Doucet ◽  
Andrew B. Duncan ◽  
Paul Fearnhead ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Markov Processes ◽  
Piecewise Deterministic Markov Processes ◽  
Restricted Domains

scholarly journals Thinning and multilevel Monte Carlo methods for piecewise deterministic (Markov) processes with an application to a stochastic Morris–Lecar model

2020 ◽  
Vol 52 (1) ◽  
pp. 138-172
Author(s):  
Vincent Lemaire ◽  
MichÉle Thieullen ◽  
Nicolas Thomas
Keyword(s):  
Monte Carlo ◽  
Markov Processes ◽  
Building Blocks ◽  
Not Given ◽  
Multilevel Monte Carlo ◽  
The Third ◽  
Piecewise Deterministic Markov Processes ◽  
Deterministic Processes ◽  
Piecewise Deterministic Processes ◽  
Weak Error

AbstractIn the first part of this paper we study approximations of trajectories of piecewise deterministic processes (PDPs) when the flow is not given explicitly by the thinning method. We also establish a strong error estimate for PDPs as well as a weak error expansion for piecewise deterministic Markov processes (PDMPs). These estimates are the building blocks of the multilevel Monte Carlo (MLMC) method, which we study in the second part. The coupling required by the MLMC is based on the thinning procedure. In the third part we apply these results to a two-dimensional Morris–Lecar model with stochastic ion channels. In the range of our simulations the MLMC estimator outperforms classical Monte Carlo.

scholarly journals Modelling and numerical assessment of a maintenance strategy with stock through piecewise deterministic Markov processes and quasi Monte Carlo methods

2017 ◽  
Vol 231 (4) ◽  
pp. 429-445
Author(s):  
Jeanne Demgne ◽  
Sophie Mercier ◽  
William Lair ◽  
Jérôme Lonchampt
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Markov Processes ◽  
Net Present Value ◽  
Spare Parts ◽  
Present Value ◽  
Quasi Monte Carlo ◽  
The Monte Carlo Method ◽  
Piecewise Deterministic Markov Processes ◽  
The Right

To ensure a power generation level, the French national electricity supply (EDF) has to manage its producing assets by putting in place adapted preventive maintenance strategies. In this article, a fleet of identical components is considered, which are spread out all around France (one per power plant site). The components are assumed to have stochastically independent lifetimes, but they are made functionally dependent through the sharing of a common stock of spare parts. When available, these spare parts are used for both corrective and preventive replacements, with priority to corrective replacements. When the stock is empty, replacements are delayed until the arrival of new spare parts. These spare parts are expensive, and their manufacturing time is long, which makes it necessary to rigorously define their ordering process. The point of the article is to provide the decision maker with the tools to take the right decision (make or not the overhaul). To do that, two indicators are proposed, which are based on an economic variable called the net present value. The net present value stands for the difference between the cumulated discounted cash-flows of the purely corrective policy and the preventive one which including the overhaul. Piecewise deterministic Markov processes are first considered for the joint modelling of the stochastic evolution of the components, stock and ordering process with and without overhaul. The indicators are next expressed with respect to these piecewise deterministic Markov processes, which have to be numerically assessed. Instead of using the most classical Monte Carlo simulations, we here suggest alternate methods based on quasi Monte Carlo simulations, which replace the random uniform numbers of the Monte Carlo method by deterministic sequences called low-discrepancy sequences. The obtained results show a real gain of the quasi Monte Carlo methods in comparison with the Monte Carlo method. The developed tools can hence help the decision maker to take the right decision.

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