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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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References

Anderson, D. F. andHigham, D. J. (2012). Multilevel Monte Carlo for continuous time Markov chains, with applications in biochemical kinetics. Multiscale Model. Simul. 10, 146–179.CrossRefGoogle Scholar
Anderson, D. F., Higham, D. J. andSun, Y. (2014). Complexity of multilevel Monte Carlo tau-leaping. SIAM J. Numer. Anal. 52, 3106–3127.CrossRefGoogle Scholar
Benam, M., Le Borgne, S., Malrieu, F. andZitt, P.-A. (2012). Quantitative ergodicity for some switched dynamical systems. Electron. Commun. Probab. 17, 1–14, 56.Google Scholar
Brémaud, P. (1981). Point Processes and Queues, Martingale Dynamics. Springer, New York.CrossRefGoogle Scholar
Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models. J. R. Statist. Soc. 46, 353–388.Google Scholar
Davis, M. H. A. (1993). Markov Models and Optimization. Chapman & Hall, London.CrossRefGoogle Scholar
Dereich, S. (2011). Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction. J. Appl. Prob. 21, 283–311.CrossRefGoogle Scholar
Dereich, S. andHeidenreich, F. (2011). A multilevel Monte Carlo algorithm for Lévy-driven stochastic differential equations. Stoch. Process. Appl. 121, 1565–1587.CrossRefGoogle Scholar
Devroye, L. (1986). Non-Uniform Random Variate Generation. Springer, New York.CrossRefGoogle Scholar
Ferreiro-Castilla, A., Kyprianou, A. E., Scheichl, R. andSuryanarayana, G. (2014). Multilevel Monte Carlo simulation for Lévy processes based on the Wiener–Hopf factorisation. Stoch. Process. Appl. 124, 985–1010.CrossRefGoogle Scholar
Giles, M. B. (2008). Multilevel Monte Carlo path simulation. Operat. Res. 56, 607–617.CrossRefGoogle Scholar
Giles, M. B. (2015). Multilevel Monte Carlo methods. In Acta Numerica 24, pp. 259–328. Cambridge University Press.CrossRefGoogle Scholar
Giorgi, D. (2017). Théorèmes limites pour estimateurs Multilevel avec et sans poids: comparaisons et applications. Doctoral thesis, Université Pierre et Marie Curie – Paris 6.Google Scholar
Glynn, P. W. andRhee, C.-H. (2014). Exact estimation for Markov chain equilibrium expectations. J. Appl. Prob. 51, 377–389.CrossRefGoogle Scholar
Glynn, P. W. andRhee, C.-H. (2015). Unbiased estimation with square root convergence for SDE models. Operat. Res. 63, 1026–1043.Google Scholar
Graham, C. andTalay, D. (2013). Stochastic Simulation and Monte Carlo. Springer.CrossRefGoogle Scholar
Hairer, E., Nørsett, S. P. andWanner, G. (2008). Solving Ordinary Differential Equations I, 2nd revised edn. Springer.Google Scholar
Heinrich, S. (2001). Multilevel Monte Carlo methods. In Large-Scale Scientific Computing (Lecture Notes Comput. Sci. 2179), eds S. Margenov et al., pp. 58–67. Springer, Berlin and Heidelberg.Google Scholar
Hodgkin, A. L. andHuxley, A. F. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. 117, 500–544.CrossRefGoogle ScholarPubMed
Jacobsen, M. (2006). Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes. Birkhäuser, Boston.Google Scholar
Lemaire, V. andPagés, G. (2017). Multilevel Richardson–Romberg extrapolation. Bernoulli 23 (4A), 2643–2692.CrossRefGoogle Scholar
Lemaire, V., Thieullen, M. andThomas, N. (2018). Exact simulation of the jump times of a class of piecewise deterministic Markov processes. J. Sci. Comput. 75, 1776–1807.CrossRefGoogle Scholar
Morris, C. andLecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophys. J. 35, 193–213.CrossRefGoogle ScholarPubMed
Pagés, G. (2018). Numerical Probability: An Introduction with Applications to Finance (Universitext). Springer, Cham.CrossRefGoogle Scholar
Pakdaman, K., Thieullen, M. andWainrib, G. (2010). Fluid limit theorems for stochastic hybrid systems with application to neuron models. Adv. Appl. Prob. 42 (3), 761–794.CrossRefGoogle Scholar
Palmowski, Z. andRolski, T. (2002). A technique for exponential change of measure for Markov processes. Bernoulli 8 (6), 767–785.Google Scholar
Talay, D. andTubaro, L. (1990). Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8 (4), 483–509.CrossRefGoogle Scholar
Xia, Y. andGiles, M. B. (2012). Multilevel path simulation for jump-diffusion SDEs. In Monte Carlo and Quasi-Monte Carlo Methods 2010 (Springer Proc. Math. Statist. 23), eds L. Plaskota and H. Woźniakowski, pp. 695–708. Springer, Berlin and Heidelberg.CrossRefGoogle Scholar