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

Part of: Markov processes Stochastic processes Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  29 April 2020

Vincent Lemaire ,
MichÉle Thieullen  and
Nicolas Thomas
Vincent Lemaire*
Affiliation:
Sorbonne University
MichÉle Thieullen*
Affiliation:
Sorbonne University
Nicolas Thomas*
Affiliation:
Sorbonne University
*
*Postal address: Laboratory of Probability, Statistics and Modeling (LPSM), UMR CNRS 8001, Sorbonne University–Campus Pierre et Marie Curie, Case 158, 4 place Jussieu, F-75252 Paris Cedex 5, France. Email addresses: vincent.lemaire@sorbonne-universite.fr, michele.thieullen@upmc.fr, nicolas.thomas@upmc.fr
*Postal address: Laboratory of Probability, Statistics and Modeling (LPSM), UMR CNRS 8001, Sorbonne University–Campus Pierre et Marie Curie, Case 158, 4 place Jussieu, F-75252 Paris Cedex 5, France. Email addresses: vincent.lemaire@sorbonne-universite.fr, michele.thieullen@upmc.fr, nicolas.thomas@upmc.fr
*Postal address: Laboratory of Probability, Statistics and Modeling (LPSM), UMR CNRS 8001, Sorbonne University–Campus Pierre et Marie Curie, Case 158, 4 place Jussieu, F-75252 Paris Cedex 5, France. Email addresses: vincent.lemaire@sorbonne-universite.fr, michele.thieullen@upmc.fr, nicolas.thomas@upmc.fr
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Abstract

In 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.

Keywords

Piecewise deterministic (Markov) processesmultilevel Monte Carlothinningstrong error estimateweak error expansionMorris–Lecar model

MSC classification

Primary: 65C05: Monte Carlo methods 65C20: Models, numerical methods
Secondary: 60G55: Point processes 60J25: Continuous-time Markov processes on general state spaces 68U20: Simulation
Type
Original Article
Information
Advances in Applied Probability , Volume 52 , Issue 1 , March 2020 , pp. 138 - 172
Copyright
© Applied Probability Trust 2020

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