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A method to compute the transition function of a piecewise deterministic Markov process with application to reliability

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  • Chiquet, Julien
  • Limnios, Nikolaos

Abstract

We study the time evolution of an increasing stochastic process governed by a first-order stochastic differential system. This defines a particular piecewise deterministic Markov process (PDMP). We consider a Markov renewal process (MRP) associated to the PDMP and its Markov renewal equation (MRE) which is solved in order to obtain a closed-form solution of the transition function of the PDMP. It is then applied in the framework of survival analysis to evaluate the reliability function of a given system. We give a numerical illustration and we compare this analytical solution with the Monte Carlo estimator.

Suggested Citation

  • Chiquet, Julien & Limnios, Nikolaos, 2008. "A method to compute the transition function of a piecewise deterministic Markov process with application to reliability," Statistics & Probability Letters, Elsevier, vol. 78(12), pages 1397-1403, September.
    • Handle: RePEc:eee:stapro:v:78:y:2008:i:12:p:1397-1403
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    References listed on IDEAS

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    1. Julien Chiquet & Nikolaos Limnios, 2006. "Estimating Stochastic Dynamical Systems Driven by a Continuous-Time Jump Markov Process," Methodology and Computing in Applied Probability, Springer, vol. 8(4), pages 431-447, December.
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    Cited by:

    1. Romain Azaïs & Alexandre Genadot, 2015. "Semi-parametric inference for the absorption features of a growth-fragmentation model," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 341-360, June.
    2. Robert Eymard & Sophie Mercier & Michel Roussignol, 2011. "Importance and Sensitivity Analysis in Dynamic Reliability," Methodology and Computing in Applied Probability, Springer, vol. 13(1), pages 75-104, March.
    3. Jiang, Shan & Li, Yan-Fu, 2021. "Dynamic Reliability Assessment of Multi-cracked Structure under Fatigue Loading via Multi-State Physics Model," Reliability Engineering and System Safety, Elsevier, vol. 213(C).
    4. Romain Azaïs & François Dufour & Anne Gégout-Petit, 2014. "Non-Parametric Estimation of the Conditional Distribution of the Interjumping Times for Piecewise-Deterministic Markov Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(4), pages 950-969, December.

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