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

2012 ◽  
Vol 44 (1) ◽  
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.

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.

Lumpability for non-irreducible finite markov chains

1984 ◽  
Vol 21 (03) ◽  
pp. 567-574 ◽  
Author(s):  
Atef M. Abdel-Moneim ◽  
Frederick W. Leysieffer
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Discrete Time ◽  
The State ◽  
Discrete Time Markov Chain ◽  
Finite Markov Chains

Conditions under which a function of a finite, discrete-time Markov chain, X(t), is again Markov are given, when X(t) is not irreducible. These conditions are given in terms of an interrelationship between two partitions of the state space of X(t), the partition induced by the minimal essential classes of X(t) and the partition with respect to which lumping is to be considered.

Time reversal and age distributions, I. Discrete-time Markov chains

1980 ◽  
Vol 17 (1) ◽  
pp. 33-46 ◽  
Author(s):  
S. Tavaré
Keyword(s):  
Population Genetics ◽  
Markov Chain ◽  
Markov Chains ◽  
Discrete Time ◽  
Time Reversal ◽  
Age Distribution ◽  
Applied Probability ◽  
Discrete Time Markov Chain

The connection between the age distribution of a discrete-time Markov chain and a certain time-reversed Markov chain is exhibited. A method for finding properties of age distributions follows simply from this approach. The results, which have application in several areas in applied probability, are illustrated by examples from population genetics.

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