Bayesian Estimation for the Markov-Modulated Diffusion Risk Model

  • F. Baltazar-LariosEmail author
  • Luz Judith R. Esparza
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 301)


We consider the Markov-modulated diffusion risk model in which the claim inter-arrivals, claim sizes, premiums, and volatility diffusion process are influenced by an underlying Markov jump process. We propose a method for obtaining the maximum likelihood estimators of its parameters using a Markov chain Monte Carlo algorithm. We present simulation studies to estimate the ruin probability in finite time using the estimators obtained with the method proposed in this paper.


Ruin probability Bayesian estimation Markov-modulated 



Luz Judith Rodriguez Esparza is supported by a Catedra CONACyT. The research of F. Baltazar-Larios was supported by PAPIIT-IA105716. Both authors are thankful to the reviewers for their invaluable comments and suggestions, which improve the paper substantially.


  1. 1.
    Asmussen, S.: Stationary distributions via first passage times. In: Dshalalow, J.H. (ed.) Advances in queueing: Theory, methods, and open problems, pp. 79–102. CRC Press, Boca Raton (1995)Google Scholar
  2. 2.
    Asmussen, S., Hobolth, A.. Markov bridges, bisection and variance reduction. In: Plaskota, L., Wozniakowski, H. (eds.), Monte Carlo and Quasi-Monte Carlo Methods 2010, Springer Proceedings in Mathematics and Statistics, vol. 23, p. 322. Springer, Berlin (2012)CrossRefGoogle Scholar
  3. 3.
    Bäuerle, N., Kötter, M.: Markov-modulated diffusion risk models. Scand. Actuar. J. 1, 34–52 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bladt, M., Sorensen, M.: Statistical inference for discretely observed markov jump processes. J. R. Stat. Soc. 67(3), 395–410 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gershmana, S.J., Blei, D.M.: A tutorial on Bayesian nonparametric models. J. Math. Psychol. 56, 1–12 (2012)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Guillou, A., Loisel, S., Stupfler, G.: Estimation of the parameters of a Markov-modulated loss process in insurance. Insur. Math. Econom. 53, 388–404 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Guillou, A., Loisel, S., Stupfler, G.: Estimating the parameters of a seasonal Markov-modulated Poisson process. Stat. Methodol. 26, 103–123 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    James, G., Witten, D., Hastie, T., Tibshirani, R.: An Introduction to Statistical Learning with Applications in R. Springer, New York (2014)zbMATHGoogle Scholar
  9. 9.
    Lu, Y., Li, S.: On the probability of ruin in a Markov-modulated risk model. Insur.: Math. Econ. Elsevier. 37(3), pp. 522–532 (2005)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Ng, A.C.Y., Yang, H.: On the joint distribution of surplus before and after ruin under a Markovian regime switching model. Stoch. Process. Their Appl. 116(2), 244–266 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Roberts, G.O., Stramer, O.: On inference for partially observed nonlinear diffusion models using the metropolis-hastings algorithm. Biometrika Trust. 88(3), 603–621 (2001)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Stanford, D.A., Yu, K., Ren, J.: Erlangian approximation to finite time ruin probabilities in perturbed risk models. Scand. Actuar. J. 1, 38–58 (2011)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ward, J.H.: Hierarchical grouping to optimize an objective function. J. Am. Stat. Assoc. 58(301), 236–244 (1963)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Facultad de CienciasUniversidad Nacional Autónoma de MéxicoMexico CityMexico
  2. 2.Catedra CONACyTUniversidad Autonoma ChapingoTexcocoMexico

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