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Improved Asymptotics for Ruin Probabilities

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Modern Problems in Insurance Mathematics

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Abstract

This chapter presents a survey of results on improved asymptotics for ruin probabilities in the Cramér–Lundberg, diffusion, and stable approximations of ruin probabilities for perturbed risk processes, obtained by the author and his collaborators. These results are: exponential asymptotic expansions for ruin probabilities in the Cramér–Lundberg and diffusion approximations of ruin probabilities; necessary and sufficient conditions for convergence of ruin probabilities in the model of diffusion and stable approximations; and explicit exponential rates of convergence in the Cramér–Lundberg approximation for ruin probabilities for reinsurance risk processes.

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References

  1. Anisimov, V.V.: Switching Processes in Queuing Models. Applied Stochastic Models Series. ISTE, Washington (2007)

    Google Scholar 

  2. Asmussen, S.: Ruin Probabilities. Advanced Series on Statistical Science and Applied Probability, vol. 2, 2nd edn. World Scientific, Singapore (2000, 2010)

    Google Scholar 

  3. Bening, V.E., Korolev, V.Yu.: Generalized Poisson Models and their Applications in Insurance and Finance. Modern Probability and Statistics. VSP, Utrecht (2002)

    Google Scholar 

  4. Bernarda, R., Vallois, P., Volpi, A., Bernarda, R., Vallois, P.: Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes. ESAIM: Probab. Statist. 12, 58–93 (2008)

    Article  Google Scholar 

  5. Blanchet, J., Zwart, B.: Asymptotic expansions of defective renewal equations with applications to perturbed risk models and processors sharing queues. Math. Methods Oper. Res. 72(2), 311–326 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bühlmann, H.: Mathematical Methods in Risk Theory. Fundamental Principles of Mathematical Sciences, vol. 172. Springer, Berlin (1970, 1996)

    Google Scholar 

  7. Cramér, H.: Review of F. Lundberg. Skand Aktuarietidskr. 9, 223–245 (1926)

    Google Scholar 

  8. Cramér, H.: On the Mathematical Theory of Risk. Skandia Jubilee Volume, Stockholm (1930)

    Google Scholar 

  9. Cramér, H.: Collective Risk Theory. Skandia Jubilee Volume, Stockholm (1955)

    Google Scholar 

  10. de Vylder, F.: Advanced Risk Theory. A Self-Contained Introduction. Editions de l’Université de Bruxelles and Swiss Association of Actuaries, Zürich (1996)

    Google Scholar 

  11. Dickson, D.C.M.: Insurance Risk and Ruin. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  12. Drozdenko, M.: Weak convergence of first-rare-event times for semi-Markov processes I. Theory Stoch. Process 13(4), 29–63 (2007a)

    MathSciNet  Google Scholar 

  13. Drozdenko, M.: Weak convergence of first-rare-event times for semi-Markov processes. Doctoral Dissertation, vol. 49. Mälardalen University, Västerås (2007b)

    Google Scholar 

  14. Drozdenko, M.: Weak convergence of first-rare-event times for semi-Markov processes II. Theory Stoch. Process 15(2), 99–118 (2009)

    MATH  MathSciNet  Google Scholar 

  15. Ekheden, E., Silvestrov, D.: Coupling and explicit rates of convergence in cramér-lundberg approximation for reinsurance risk processes. Comm. Statist. Theor. Methods 40, 3524–3539 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  16. Embrechts, P., Klüppelberg, C.: Modelling Extremal Events for Insurance and Finance. Applications of Mathematics, vol. 33. Springer, Berlin (1997)

    Google Scholar 

  17. Englund, E.: Nonlinearly perturbed renewal equations with applications to a random walk. Theory Stoch. Process 6(22), 3–4, 33–60 (2000)

    Google Scholar 

  18. Englund, E.: Nonlinearly perturbed renewal equations with applications. Doctoral Dissertation, Umeå University (2001)

    Google Scholar 

  19. Englund, E., Silvestrov, D.S.: Mixed large deviation and ergodic theorems for regenerative processes with discrete time. In: Jagers, P., Kulldorff, G., Portenko, N., Silvestrov, D. (eds.) Proceedings of the Second Scandinavian-Ukrainian Conference in Mathematical Statistics, vol. I, Umeå (1997). (Theory Stochastic Process 3(19), no. 1–2, 164–176, 1997)

    Google Scholar 

  20. Feller, W.: An Introduction to Probability Theory and its Applications, vol. II. Wiley Series in Probability and Statistics. Wiley, New York (1966, 1971)

    Google Scholar 

  21. Gerber, H.U.: An Introduction to Mathematical Risk Theory. Huebner Foundation Monographs, Philadelphia (1979)

    Google Scholar 

  22. Grandell, J.: Aspects of Risk Theory. Probability and Its Applications. Springer, New York (1991)

    Book  MATH  Google Scholar 

  23. Gyllenberg, M., Silvestrov, D.S.: Cramér-lundberg and diffusion approximations for nonlinearly perturbed risk processes including numerical computation of ruin probabilities. Theory Stoch. Process 5(21), 1–2, 6–21 (1999)

    Google Scholar 

  24. Gyllenberg, M., Silvestrov, D.S.: Nonlinearly perturbed regenerative processes and pseudo-stationary phenomena for stochastic systems. Stoch. Process. Appl. 86, 1–27 (2000a)

    Article  MATH  MathSciNet  Google Scholar 

  25. Gyllenberg, M., Silvestrov, D.S.: Cramér-lundberg approximation for nonlinearly perturbed risk processes. Insur. Math. Econ. 26, 75–90 (2000b)

    Article  MATH  MathSciNet  Google Scholar 

  26. Gyllenberg, M., Silvestrov, D.S.: Quasi-Stationary Phenomena in Nonlinearly Perturbed Stochastic Systems. De Gruyter Expositions in Mathematics, vol. 44. Walter de Gruyter, Berlin (2008)

    Google Scholar 

  27. Iglehart, D.L.: Diffusion approximations in collective risk theory. J. Appl. Probab. 6, 285–292 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  28. Kalashnikov, V.V.: Geometric Sums: Bounds for Rare Events with Applications. Mathematics and Its Applications, vol. 413. Kluwer, Dordrecht (1997)

    Google Scholar 

  29. Kartashov, M.V.: Strong Stable Markov Chains. VSP, Utrecht and TBiMC, Kiev (1996a)

    MATH  Google Scholar 

  30. Kartashov, M.V.: Computation and estimation of the exponential ergodicity exponent for general Markov processes and chains with recurrent kernels. Teor. \(\check{\rm {I}}\)movirn. Mat. Stat. 54, 47–57 (1996b). (English translation in Theory Probab. Math. Statist. 54, 49–60)

    Google Scholar 

  31. Kartashov, M.V., Golomoziy, V.V.: Maximal coupling and stability of discrete Markov chains. I. Teor. Imovirn. Mat. Stat. 86, 81–91 (2011)

    Google Scholar 

  32. Korolyuk, D.V., Silvestrov, D.S.: Entry times into asymptotically receding domains for ergodic Markov chains. Teor. Veroyatn. Primen. 28, 410–420 (1983). (English translation in Theory Probab. Appl. 28, 432–442)

    Google Scholar 

  33. Korolyuk, D.V., Silvestrov, D.S.: Entry times into asymptotically receding regions for processes with semi-Markov switchings. Teor. Veroyatn. Primen. 29, 539–544 (1984). (English translation in Theory Probab. Appl. 29, 558–563)

    Google Scholar 

  34. Koroliuk, V.S., Limnios, N.: Stochastic Systems in Merging Phase Space. World Scientific, Singapore (2005)

    Book  MATH  Google Scholar 

  35. Kovalenko, I.N.: On the class of limit distributions for thinning flows of homogeneous events. Litov. Mat. Sb. 5, 569–573 (1965)

    MATH  MathSciNet  Google Scholar 

  36. Kruglov, V.M., Korolev, VYu.: Limit Theorems for Random Sums. Izdatel’stvo Moskovskogo Universiteta, Moscow (1990)

    MATH  Google Scholar 

  37. Lindvall, T.: Lectures on the Coupling Method. Wiley Series in Probability and Mathematical Statistics. Wiley, New York and Dover, Mineola (1992, 2002)

    Google Scholar 

  38. Lundberg, F.: I. Approximerad framställning av sannolikhetsfunktionen. II. Återförsäkring av kollektivrisker. Almqvist & Wiksell, Uppsala (1903)

    Google Scholar 

  39. Lundberg, F. (1909). Über die der Theorie Rückversicherung. In: VI Internationaler Kongress für Versicherungswissenschaft. Bd. 1, Vien, pp. 877–955 (1909)

    Google Scholar 

  40. Lundberg, F.: Försäkringsteknisk Riskutjämning. F. Englunds boktryckeri AB, Stockholm (1926)

    Google Scholar 

  41. Mikosch, T.: Non-life Insurance Mathematics. An Introduction with Stochastic Processes. Springer, Berlin (2004)

    MATH  Google Scholar 

  42. Ni, Y.: Perturbed renewal equations with multivariate nonpolynomial perturbations. In: Frenkel, I., Gertsbakh, I., Khvatskin, L., Laslo, Z., Lisnianski, A. (eds.) Proceedings of the International Symposium on Stochastic Models in Reliability Engineering, pp. 754–763. Life Science and Operations Management, Beer Sheva, Israel (2010a)

    Google Scholar 

  43. Ni, Y.: Analytical and numerical studies of perturbed renewal equations with multivariate non-polynomial perturbations. J. Appl. Quant. Meth. 5(3), 498–515 (2010b)

    Google Scholar 

  44. Ni, Y.: Nonlinearly perturbed renewal equations: asymptotic results and applications. Doctoral Dissertation, vol. 106. Mälardalen University, Västerås (2011)

    Google Scholar 

  45. Ni, Y.: Nonlinearly perturbed renewal equations: the non-polynomial case. Teor. Imovirn. Mat. Stat. 84, 117–129 (2012)

    MATH  Google Scholar 

  46. Ni, Y.: Exponential asymptotical expansions for ruin probability in a classical risk process with non-polynomial perturbations. In: Silvestrov, D., Martin-Löf, A. (eds.) Modern Problems in Insurance Mathematics, pp. 69–93. Springer (2014)

    Google Scholar 

  47. Ni, Y., Silvestrov, D., Malyarenko, A.: Exponential asymptotics for nonlinearly perturbed renewal equation with non-polynomial perturbations. J. Numer. Appl. Math. 1(96), 173–197 (2008)

    Google Scholar 

  48. Petersson, M.: Quasi-stationary distributions for perturbed discrete time regenerative processes. Teor. Imovirn. Mat. Stat. 89 (2014, forthcoming)

    Google Scholar 

  49. Petersson, M.: Asymptotics of ruin probabilities for perturbed discrete time risk processes. In: Silvestrov, D., Martin-Löf, A. (eds.) Modern Problems in Insurance Mathematics, pp. 95–112. Springer (2014)

    Google Scholar 

  50. Rolski, T., Schmidli, H., Schmidt, V., Teugels, J.: Stochastic Processes for Insurance and Finance. Wiley Series in Probability and Statistics. Wiley, New York (1999)

    Book  MATH  Google Scholar 

  51. Schmidli, H.: A General Insurance Risk Model. Dr. Sc. Thesis, ETH, Zurich (1992)

    Google Scholar 

  52. Schmidli, H.: Characteristics of ruin probabilities in classical risk models with and without investment, Cox risk models and perturbed risk models. Dissertation, University of Aarhus, Aarhus, Memoirs, vol. 15. University of Aarhus, Department of Theoretical Statistics, Aarhus (2000)

    Google Scholar 

  53. Shurenkov, V.M.: Transition phenomena of the renewal theory in asymptotical problems of theory of random processes 1. Mat. Sb. 112, 115–132 (1980a). (English translation in Math. USSR Sbornik 40, no. 1, 107–123)

    Google Scholar 

  54. Shurenkov, V.M.: Transition phenomena of the renewal theory in asymptotical problems of theory of random processes 2. Mat. Sbornik 112, 226–241 (1980b). (English translation in Math. USSR Sbornik 40, no. 2, 211–225)

    Google Scholar 

  55. Silvestrov, D.S.: Limit Theorems for Composite Random Functions. Izdatel’stvo “Vysca Scola” and Izdatel’stvo Kievskogo Universiteta, Kiev (1974)

    Google Scholar 

  56. Silvestrov, D.S.: A generalization of the renewal theorem. Dokl. Akad. Nauk Ukr. SSR. Ser. A 11, 978–982 (1976)

    Google Scholar 

  57. Silvestrov, D.S.: The renewal theorem in a series scheme. 1. Teor. Veroyatn. Mat. Stat. 18, 144–161 (1978). (English translation in Theory Probab. Math. Stat. 18, 155–172)

    Google Scholar 

  58. Silvestrov, D.S.: The renewal theorem in a series scheme 2. Teor. Veroyatn. Mat. Stat. 20, 97–116 (1979). (English translation in Theory Probab. Math. Stat. 20, 113–130)

    Google Scholar 

  59. Silvestrov, D.S.: Semi-Markov Processes with a Discrete State Space. Library for the Engineer in Reliability. Sovetskoe Radio, Moscow (1980a)

    Google Scholar 

  60. Silvestrov, D.S.: Synchronized regenerative processes and explicit estimates for the rate of convergence in ergodic theorems. Dokl. Acad. Nauk Ukr. SSR Ser. A 11, 22–25 (1980b)

    Google Scholar 

  61. Silvestrov, D.S.: Explicit estimates in ergodic theorems for regenerative processes. Elektron. Inf. Kybern. 16, 461–463 (1980c)

    MathSciNet  Google Scholar 

  62. Silvestrov, D.S.: Method of a single probability space in ergodic theorems for regenerative processes 1. Math. Oper. Stat. Ser. Optim. 14, 285–299 (1983)

    MathSciNet  Google Scholar 

  63. Silvestrov, D.S.: Method of a single probability space in ergodic theorems for regenerative processes 2. Math. Oper. Stat. Ser. Optim. 15, 601–612 (1984a)

    MathSciNet  Google Scholar 

  64. Silvestrov, D.S.: Method of a single probability space in ergodic theorems for regenerative processes 3. Math. Oper. Stat. Ser. Optim. 15, 613–622 (1984b)

    MathSciNet  Google Scholar 

  65. Silvestrov, D.: Coupling for markov renewal processes and the rate of convergence in ergodic theorems for processes with semi-markov switchings. Acta Appl. Math. 34, 109–124 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  66. Silvestrov, D.S.: Exponential asymptotic for perturbed renewal equations. Teor. \(\check{\rm {I}}\)movirn. Mat. Stat. 52, 143–153 (1995). (English translation in Theory Probab. Math. Stat. 52, 153–162)

    Google Scholar 

  67. Silvestrov, D.S.: Perturbed renewal equation and diffusion type approximation for risk processes. Teor. \(\check{\rm {I}}\)movirn. Mat. Stat. 62, 134–144 (2000). (English translation in Theory Probab. Math. Stat. 62, 145–156)

    Google Scholar 

  68. Silvestrov, D.S.: Limit Theorems for Randomly Stopped Stochastic Processes. Probability and its Applications. Springer, London (2004)

    Book  MATH  Google Scholar 

  69. Silvestrov D.S.: Nonlinearly perturbed stochastic processes and systems. In: Rykov, V., Balakrishnan, N., Nikulin, M. (eds.) Mathematical and Statistical Models and Methods in Reliability, Birkhäuser, Chapter 2, 19–38 (2010)

    Google Scholar 

  70. Silvestrov, D.S., Drozdenko, M.O.: Necessary and sufficient conditions for the weak convergence of the first-rare-event times for semi-Markov processes. Dopov. Nac. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki 11, 25–28 (2005)

    Google Scholar 

  71. Silvestrov, D.S., Drozdenko, M.O.: Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes I. Theory Stoch. Process 12(28), 3–4, 187–202 (2006a)

    Google Scholar 

  72. Silvestrov, D.S.: Asymptotic expansions for distributions of the surplus prior and at the time of ruin. Theory Stoch. Process. 13(29), no. 4, 183–188 (2007)

    Google Scholar 

  73. Silvestrov, D.S., Drozdenko, M.O.: Necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes II. Theory Stoch. Proces. 12(28), 3–4, 187–202 (2006b)

    Google Scholar 

  74. Silvestrov, D.S., Petersson, M.: Exponential expansions for perturbed discrete time renewal equations. In: Karagrigoriou, A., Lisnianski, A., Kleyner, A., Frenkel, I. (eds.) Applied Reliability Engineering and Risk Analysis. Probabilistic Models and Statistical Inference, Chapter 23, pp. 349–362. Wiley, Chichester (2014)

    Google Scholar 

  75. Silvestrov, D.S., Velikii, A.Yu.: Stability problems for stochastic models. In: Zolotarev, V.M., Kalashnikov, V.V. (eds.) Necessary and sufficient conditions for convergence of attainment times, pp. 129–137. Trudy Seminara, VNIISI, Moscow (1988). (English translation in J. Soviet. Math. 57, 3317–3324)

    Google Scholar 

  76. Thorisson, H.: Coupling. Stationarity and Regeneration. Probability and Its Applications. Springer, New York (2000)

    MATH  Google Scholar 

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  1. Stockholm University, Stockholm, Sweden

    Dmitrii Silvestrov

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  1. Department of Mathematics Professor of Mathematical Statistics, Stockholm University, Stockholm, Sweden

    Dmitrii Silvestrov

  2. Department of Mathematics Professor in Mathematical Statistics, Stockholm University, Stockholm, Sweden

    Anders Martin-Löf

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Silvestrov, D. (2014). Improved Asymptotics for Ruin Probabilities. In: Silvestrov, D., Martin-Löf, A. (eds) Modern Problems in Insurance Mathematics. EAA Series. Springer, Cham. https://doi.org/10.1007/978-3-319-06653-0_5

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