scholarly journals A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process

2014 ◽  
Vol 18 ◽  
pp. 726-749 ◽  
Author(s):  
Romain Azaïs
Keyword(s):  
Markov Process ◽  
Transition Kernel ◽  
Nonparametric Estimator ◽  
Piecewise Deterministic Markov Process

scholarly journals Averaging for a Fully Coupled Piecewise-Deterministic Markov Process in Infinite Dimensions

2012 ◽  
Vol 44 (3) ◽  
pp. 749-773 ◽  
Author(s):  
Alexandre Genadot ◽  
Michèle Thieullen
Keyword(s):  
Ion Channels ◽  
Markov Process ◽  
Time Scale ◽  
Evolution Equations ◽  
Scale Model ◽  
Infinite Dimensions ◽  
Fast Analysis ◽  
Piecewise Deterministic Markov Process ◽  
Averaged Model ◽  
Fully Coupled

In this paper we consider the generalized Hodgkin-Huxley model introduced in Austin (2008). This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully coupled piecewise-deterministic Markov process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that, asymptotically, this ‘two-time-scale’ model reduces to the so-called averaged model, which is still a PDMP in infinite dimensions, for which we provide effective evolution equations and jump rates.

Global dynamical behavior of a multigroup SVIR epidemic model with Markovian switching

2021 ◽  
pp. 2150080
Author(s):  
Qun Liu ◽  
Daqing Jiang
Keyword(s):  
Markov Process ◽  
Lyapunov Function ◽  
Stochastic System ◽  
Epidemic Model ◽  
Regime Switching ◽  
Dynamical Behavior ◽  
Piecewise Deterministic Markov Process ◽  
Positive Recurrence ◽  
Persistence In The Mean ◽  
The Mean

In this paper, we are concerned with the global dynamical behavior of a multigroup SVIR epidemic model, which is formulated as a piecewise-deterministic Markov process. We first obtain sufficient criteria for extinction of the diseases. Then we establish sufficient criteria for persistence in the mean of the diseases. Moreover, in the case of persistence, we find a domain which is positive recurrence for the solution of the stochastic system by constructing an appropriate Lyapunov function with regime switching.

scholarly journals Averaging for a Fully Coupled Piecewise-Deterministic Markov Process in Infinite Dimensions

2012 ◽  
Vol 44 (03) ◽  
pp. 749-773 ◽  
Author(s):  
Alexandre Genadot ◽  
Michèle Thieullen
Keyword(s):  
Ion Channels ◽  
Markov Process ◽  
Time Scale ◽  
Evolution Equations ◽  
Scale Model ◽  
Infinite Dimensions ◽  
Fast Analysis ◽  
Piecewise Deterministic Markov Process ◽  
Averaged Model ◽  
Fully Coupled

In this paper we consider the generalized Hodgkin-Huxley model introduced in Austin (2008). This model describes the propagation of an action potential along the axon of a neuron at the scale of ion channels. Mathematically, this model is a fully coupled piecewise-deterministic Markov process (PDMP) in infinite dimensions. We introduce two time scales in this model in considering that some ion channels open and close at faster jump rates than others. We perform a slow-fast analysis of this model and prove that, asymptotically, this ‘two-time-scale’ model reduces to the so-called averaged model, which is still a PDMP in infinite dimensions, for which we provide effective evolution equations and jump rates.

scholarly journals Numerical Methods for the Exit Time of a Piecewise-Deterministic Markov Process

2012 ◽  
Vol 44 (01) ◽  
pp. 196-225 ◽  
Author(s):  
Adrien Brandejsky ◽  
Benoîte De Saporta ◽  
François Dufour
Keyword(s):  
Markov Chain ◽  
Numerical Methods ◽  
Markov Process ◽  
Numerical Method ◽  
Rate Of Convergence ◽  
Discrete Time ◽  
Exit Time ◽  
Survival Function ◽  
Piecewise Deterministic Markov Process ◽  
Discrete Time Markov Chain

We present a numerical method to compute the survival function and the moments of the exit time for a piecewise-deterministic Markov process (PDMP). Our approach is based on the quantization of an underlying discrete-time Markov chain related to the PDMP. The approximation we propose is easily computable and is even flexible with respect to the exit time we consider. We prove the convergence of the algorithm and obtain bounds for the rate of convergence in the case of the moments. We give an academic example and a model from the reliability field to illustrate the results of the paper.

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