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Numerical Solution of non-Homogeneous Semi-Markov Processes in Transient Case

Author

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  • Jacques Janssen

    (CESIAF)

  • Raimondo Manca

    (Universitá “La Sapienza” Roma)

Abstract

In this article a numerical solution for the evolution equation of a continuous time non-homogeneous semi-Markov process (NHSMP) is obtained using a quadrature method. The paper, after a short introduction to continuous time NHSMP, presents the numerical solution of the process evolution equation with a general quadrature method. Furthermore, the paper gives results that justify this approach, proving that the numerical solution tends to the evolution equation of the continuous time NHSMP. Moreover, the formulae related to some specific quadrature methods are given and a method for obtaining the discrete time NHSMP by applying a very particular quadrature formula for the discretization is shown. In this way the relation between the continuous and discrete time NHSMP is proved. Then, the problem of obtaining the continuous time NHSMP from the discrete one is considered. This problem is solved showing that the discrete process converges in law to the continuous one if the discretized time interval tends to zero. In addition, the discrete time NHSMP in matrix form is presented, and the fact that the solution to this process always exists is proved. Finally, an algorithm for solving the discrete time NHSMP is given. To illustrate the use of this algorithm for a discrete NHSMP, an example in the area of finance is presented.

Suggested Citation

  • Jacques Janssen & Raimondo Manca, 2001. "Numerical Solution of non-Homogeneous Semi-Markov Processes in Transient Case," Methodology and Computing in Applied Probability, Springer, vol. 3(3), pages 271-293, September.
    • Handle: RePEc:spr:metcap:v:3:y:2001:i:3:d:10.1023_a:1013719007075
    DOI: 10.1023/A:1013719007075
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    Citations

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    Cited by:

    1. D'Amico, Guglielmo & Guillen, Montserrat & Manca, Raimondo, 2009. "Full backward non-homogeneous semi-Markov processes for disability insurance models: A Catalunya real data application," Insurance: Mathematics and Economics, Elsevier, vol. 45(2), pages 173-179, October.
    2. Moura, Márcio das Chagas & Droguett, Enrique López, 2009. "Mathematical formulation and numerical treatment based on transition frequency densities and quadrature methods for non-homogeneous semi-Markov processes," Reliability Engineering and System Safety, Elsevier, vol. 94(2), pages 342-349.
    3. Sophie Mercier, 2008. "Numerical Bounds for Semi-Markovian Quantities and Application to Reliability," Methodology and Computing in Applied Probability, Springer, vol. 10(2), pages 179-198, June.
    4. Fuino, Michel & Wagner, Joël, 2018. "Long-term care models and dependence probability tables by acuity level: New empirical evidence from Switzerland," Insurance: Mathematics and Economics, Elsevier, vol. 81(C), pages 51-70.
    5. Maegebier, Alexander, 2013. "Valuation and risk assessment of disability insurance using a discrete time trivariate Markov renewal reward process," Insurance: Mathematics and Economics, Elsevier, vol. 53(3), pages 802-811.
    6. E. Mathieu & Y. Foucher & P. Dellamonica & J. P. Daures, 2007. "Parametric and Non Homogeneous Semi-Markov Process for HIV Control," Methodology and Computing in Applied Probability, Springer, vol. 9(3), pages 389-397, September.
    7. Guglielmo D’Amico & Raimondo Manca & Filippo Petroni & Dharmaraja Selvamuthu, 2021. "On the Computation of Some Interval Reliability Indicators for Semi-Markov Systems," Mathematics, MDPI, vol. 9(5), pages 1-23, March.

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