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Stochastic numerical methods for Piecewise Deterministic Markov Processes : applications in Neuroscience

Abstract : In this thesis, motivated by applications in Neuroscience, we study efficient Monte Carlo (MC) and Multilevel Monte Carlo (MLMC) methods based on the thinning for piecewise deterministic (Markov) processes (PDMP or PDP) that we apply to stochastic conductance-based models. On the one hand, when the deterministic motion of the PDMP is explicitly known we end up with an exact simulation. On the other hand, when the deterministic motion is not explicit, we establish strong estimates and a weak error expansion for the numerical scheme that we introduce. The thinning method is fundamental in this thesis. Beside the fact that it is intuitive, we use it both numerically (to simulate trajectories of PDMP/PDP) and theoretically (to construct the jump times and establish error estimates for PDMP/PDP).
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  • HAL Id : tel-03001270, version 3

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Nicolas Thomas. Stochastic numerical methods for Piecewise Deterministic Markov Processes : applications in Neuroscience. Probability [math.PR]. Sorbonne Université, 2019. English. ⟨NNT : 2019SORUS385⟩. ⟨tel-03001270v3⟩

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