Insurance Models Under Incomplete Information

  • Ekaterina Bulinskaya
  • Julia Gusak
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 231)


The aim of the chapter is optimization of insurance company performance under incomplete information. To this end, we consider the periodic-review model with capital injections and reinsurance studied by the authors in their previous paper for the case of known claim distribution. We investigate the stability of the one-step and multi-step model in terms of the Kantorovich metric. These results are used for obtaining almost optimal policies based on the empirical distributions of underlying processes.


Incomplete information Periodic-review insurance model Reinsurance Capital injections Optimization Stability 


  1. 1.
    Albrecher, H., Thonhauser, S.: Optimality results for dividend problems in insurance. Rev. R. Acad. Cien. Serie A. Mat. 103(2), 295–320 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Avanzi, B.: Strategies for dividend distribution: a review. N. Am. Actuar. J. 13(2), 217–251 (2009)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bulinskaya, E.: Asymptotic analysis of insurance models with bank loans. In: Bozeman, J.R., Girardin, V., Skiadas, C.H. (eds.) New Perspectives on Stochastic Modeling and Data Analysis, pp. 255–270. ISAST, Athens, Greece (2014)Google Scholar
  4. 4.
    Bulinskaya, E., Gromov, A.: Asymptotic behavior of the processes describing some insurance models. Commun. Stat. Theory Methods 45, 1778–1793 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bulinskaya, E., Gusak, J.: Optimal control and sensitivity analysis for two risk models. Commun. Stat. Simul. Comput. 44, 1–17 (2015)CrossRefGoogle Scholar
  6. 6.
    Bulinskaya, E., Muromskaya, A.: Optimization of multi-component insurance system with dividend payments. In: Manca, R., McClean, S., Skiadas, Ch.H. (eds.) New Trends in Stochastic Modeling and Data Analysis. ISAST, Athens, Greece (2015)Google Scholar
  7. 7.
    Bulinskaya, E., Sokolova, A.: Asymptotic behaviour of stochastic storage systems. Mod. Probl. Math. Mech. 10(3), 37–62 (2015) (in Russian)Google Scholar
  8. 8.
    Bulinskaya, E., Gusak, J., Muromskaya, A.: Discrete-time insurance model with capital injections and reinsurance. Methodol. Comput. Appl. Probab. 17(4), 899–914 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    De Finetti, B.: Su un’impostazione alternativa della teoria collettiva del rischio. Trans. XV-th Int. Congr. Actuar. 2, 433–443 (1957)Google Scholar
  10. 10.
    Del Barrio, E., Giné, E., Matrán, C.: Central limit theorems for the Wasserstein distance between the empirical and the true distributions. Ann. Probab. 27(2), 1009–1071 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Dickson, D.C.M., Waters, H.R.: Some optimal dividends problems. ASTIN Bulletin 34, 49–74 (2004)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Dobrushin, R.L.: Prescribing a system of random variables by conditional distributions. Theory Probab. Appl. 15(3), 458–486 (1970)CrossRefGoogle Scholar
  13. 13.
    Eisenberg, J., Schmidli, H.: Optimal control of capital injections by reinsurance in a diffusion approximation. Blätter der DGVFM 30, 1–13 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Jain, N.C.: Central limit theorems and related questions in Banah space. In: Proceedings of Symposia in Pure Mathematics, vol. 31, pp. 55–65. American Mathematical Society, Providence, RI (1977)Google Scholar
  15. 15.
    Kulenko, N., Schmidli, H.: Optimal dividend strategies in a Cramér-Lundberg model with capital injections. Insur. Math. Econ. 43, 270–278 (2008)CrossRefGoogle Scholar
  16. 16.
    Lawniczak, A.: The Levy-Lindeberg central limit theorem in Orlicz spaces. Proc. Amer. Math. Soc. 89, 673–679 (1983)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Li, S., Lu, Y., Garrido, J.: A review of discrete-time risk models. Rev. R. Acad. Cien. Serie A. Mat. 103(2), 321–337 (2009)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Oakley, J.E., O’Hagan, A.: Probabilistic sensitivity analysis of complex models: a Bayesian approach. J. R. Statist. Soc. B. 66, Part 3, 751–769 (2004)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Peskir, G., Shiryaev, A.: Optimal Stopping and Free-Boundary Problems. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel (2006)Google Scholar
  20. 20.
    Rachev, S.T., Klebanov, L., Stoyanov, S.V., Fabozzi, F.: The Methods of Distances in the Theory of Probability and Statistics. Springer, New York (2013)CrossRefGoogle Scholar
  21. 21.
    Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., Tarantola, S.: Global Sensitivity Analysis. The Primer. Wiley, New York (2008)zbMATHGoogle Scholar
  22. 22.
    Schmidli, H.: Stochastic Control in Insurance. Springer, New York (2008)zbMATHGoogle Scholar
  23. 23.
    Shorack, G.R., Wellner, J.A.: Empirical Processes with Application to Statistics. Wiley, New York (1986)zbMATHGoogle Scholar
  24. 24.
    Sobol, I.M.: Sensitivity analysis for nonlinear mathematical models. Math. Model. Comput. Expt. 1, 407–414 (1993)zbMATHGoogle Scholar
  25. 25.
    Yang, H., Gao, W., Li, J.: Asymptotic ruin probabilities for a discrete-time risk model with dependent insurance and financial risks. Scand. Actuar. J. 1, 1–17 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

Personalised recommendations