IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v37y2024i1d10.1007_s10959-023-01259-4.html
   My bibliography  Save this article

Semimartingale Representation of a Class of Semi-Markov Dynamics

Author

Listed:
  • Anindya Goswami

    (IISER Pune)

  • Subhamay Saha

    (IIT Guwahati)

  • Ravishankar Kapildev Yadav

    (IISER Pune)

Abstract

We consider a class of semi-Markov processes (SMP) such that the embedded discrete-time Markov chain may be non-homogeneous. The corresponding augmented processes are represented as semi-martingales using a stochastic integral equation involving a Poisson random measure. The existence and uniqueness of the equation are established. Subsequently, we show that the solution is indeed a SMP with desired transition rate. Finally, we derive the law of the bivariate process obtained from two solutions of the equation having two different initial conditions.

Suggested Citation

  • Anindya Goswami & Subhamay Saha & Ravishankar Kapildev Yadav, 2024. "Semimartingale Representation of a Class of Semi-Markov Dynamics," Journal of Theoretical Probability, Springer, vol. 37(1), pages 489-510, March.
    • Handle: RePEc:spr:jotpro:v:37:y:2024:i:1:d:10.1007_s10959-023-01259-4
    DOI: 10.1007/s10959-023-01259-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-023-01259-4
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-023-01259-4?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Janssen, J. & de Dominicis, R., 1984. "Finite non-homogeneous semi-Markov processes: Theoretical and computational aspects," Insurance: Mathematics and Economics, Elsevier, vol. 3(3), pages 157-165, July.
    2. 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.
    3. Milan Kumar Das & Anindya Goswami & Nimit Rana, 2016. "Risk Sensitive Portfolio Optimization in a Jump Diffusion Model with Regimes," Papers 1603.09149, arXiv.org, revised Jan 2018.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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. 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.
    3. Milan Kumar Das & Anindya Goswami, 2019. "Testing of binary regime switching models using squeeze duration analysis," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 6(01), pages 1-20, March.
    4. Aleka Papadopoulou & George Tsaklidis, 2007. "Some Reward Paths in Semi-Markov Models with Stochastic Selection of the Transition Probabilities," Methodology and Computing in Applied Probability, Springer, vol. 9(3), pages 399-411, September.
    5. Laura Eslava & Fernando Baltazar-Larios & Bor Reynoso, 2022. "Maximum Likelihood Estimation for a Markov-Modulated Jump-Diffusion Model," Papers 2211.17220, arXiv.org.
    6. 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.
    7. Milan Kumar Das & Anindya Goswami & Sharan Rajani, 2019. "Inference of Binary Regime Models with Jump Discontinuities," Papers 1910.10606, arXiv.org, revised Mar 2022.
    8. Jianmin Shi, 2025. "Superposed semi-Markov decision process with application to optimal maintenance systems," Journal of Combinatorial Optimization, Springer, vol. 49(3), pages 1-19, April.
    9. Shelby Brumelle & Darius Walczak, 2003. "Dynamic Airline Revenue Management with Multiple Semi-Markov Demand," Operations Research, INFORMS, vol. 51(1), pages 137-148, February.
    10. Lijun Bo & Huafu Liao & Xiang Yu, 2017. "Risk Sensitive Portfolio Optimization with Default Contagion and Regime-Switching," Papers 1712.05676, arXiv.org, revised Oct 2018.
    11. Milan Kumar Das & Anindya Goswami, 2018. "Testing of Binary Regime Switching Models using Squeeze Duration Analysis," Papers 1807.04393, arXiv.org, revised Aug 2018.
    12. D’Amico, Guglielmo & Petroni, Filippo & Prattico, Flavio, 2017. "Insuring wind energy production," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 467(C), pages 542-553.
    13. 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.
    14. 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.
    15. Shemendyuk, Aleksandr & Wagner, Joël, 2025. "Evolution of institutional long-term care costs based on health factors," Insurance: Mathematics and Economics, Elsevier, vol. 120(C), pages 107-130.
    16. Minglian Lin & Indranil SenGupta, 2021. "Analysis of optimal portfolio on finite and small time horizons for a stochastic volatility market model," Papers 2104.06293, arXiv.org.
    17. Marcin Pitera & Łukasz Stettner, 2023. "Discrete‐time risk sensitive portfolio optimization with proportional transaction costs," Mathematical Finance, Wiley Blackwell, vol. 33(4), pages 1287-1313, October.
    18. Guglielmo D’Amico & Fulvio Gismondi & Filippo Petroni, 2020. "Insurance Contracts for Hedging Wind Power Uncertainty," Mathematics, MDPI, vol. 8(8), pages 1-16, August.
    19. Andre Monteiro & Georgi V. Smirnov & Andre Lucas, 2006. "Nonparametric Estimation for Non-Homogeneous Semi-Markov Processes: An Application to Credit Risk," Tinbergen Institute Discussion Papers 06-024/2, Tinbergen Institute, revised 27 Mar 2006.
    20. Marcus Christiansen, 2012. "Multistate models in health insurance," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 96(2), pages 155-186, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:37:y:2024:i:1:d:10.1007_s10959-023-01259-4. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.