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Theory of Probability and Mathematical Statistics

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Inhomogeneous perturbations of a renewal equation and the Cramér-Lundberg theorem for a risk process with variable premium rates

Author: M. V. Kartashov
Translated by: O. I. Klesov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 78 (2008).
Journal: Theor. Probability and Math. Statist. 78 (2009), 61-73
MSC (2000): Primary 60J45; Secondary 60A05, 60K05
Published electronically: August 4, 2009
MathSciNet review: 2446849
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider a time inhomogeneous perturbation of the classical renewal equation with continuous time that can be reduced to the integral Volterra equation with a nonnegative bounded kernel. We assume that the kernel is approximated for large time intervals by a convolution kernel generated by a probability distribution. We prove that the limit of the solution of the perturbed equation exists if the corresponding perturbation of solutions of the perturbed equation is small.

We consider an application for ruin functions of the classical risk process where the premium rate depends on the current capital of an insurance company. We obtain the exponential asymptotic behavior with the Lundberg index evaluated from the original (nonperturbed) intensity.

References [Enhancements On Off] (What's this?)

  • 1. N. V. Kartashov, Strong Stable Markov Chains, VSP/TViMS, Utrecht/Kiev, The Netherlands/Ukraine, 1996. MR 1451375 (99e:60150)
  • 2. S. Asmussen and S. S. Petersen, Ruin probabilities expressed in terms of storage processes, Adv. Appl. Probab. 20 (1988), 913-916. MR 968005 (90f:62312)
  • 3. S. Asmussen and H. M. Nielsen, Ruin probabilities via local adjustment coefficients, J. Appl. Probab. 33 (1995), 736-755. MR 1344073 (97d:60050)
  • 4. D. C. M. Dickson and J. R. Gray, Exact solutions for ruin probability in the presence of an absorbing upper barrier, Scand. Actuar. J. 1984, 174-186. MR 781520 (86f:62178)
  • 5. D. C. M. Dickson, The probability of ultimate ruin with a variable premium rate, Scand. Actuar. J. 1991, 75-86.
  • 6. B. A. Djehiche, Large deviation estimate for ruin probabilities, Scand. Actuar. J. 1993, 42-59.
  • 7. P. Embrechts and H. Schmidli, Ruin estimation for a general insurance risk model, Adv. Appl. Probab. 26 (1994), 404-422. MR 1272719 (95g:60121)
  • 8. H. Gerber, The dilemma between dividends and safety and a generalization of the Lundberg-Cramér formulas, Scand. Actuar. J. 1974, 46-57. MR 0443874 (56:2236)
  • 9. H. Gerber, On the probability of ruin in the presence of a linear dividend barrier, Scand. Actuar. J. 1981, 105-115. MR 623405 (83c:62169)
  • 10. H. Gerber and E. S. W. Shiu, On the time value of ruin, Proc. of the 31 Actuarial Research Conference, Ball State Univ., Aug. 1996, pp. 145-199.
  • 11. S. S. Petersen, Calculation of ruin probabilities when the premium depends on the current reserve, Scand. Actuar. J. 1990, 147-159. MR 1067100 (91f:62166)
  • 12. P. Picard and C. Lefevre, On the first crossing of the surplus process with a given upper barrier, Insurance Math. Econom. 14 (1994), 163-179. MR 1292961
  • 13. H. Schmidli, An extension to the renewal theorem and an application to risk theory, Ann. Appl. Probab. 7 (1997), no. 1, 121-133. MR 1428752 (97k:60234)
  • 14. G. C. Tailor, Probability of ruin with variable premium rate, Scand. Actuar. J. 1980, 57-76. MR 578447 (81m:62188)
  • 15. N. V. Kartashov, On ruin probabilities for a risk process with bounded reserves, Teor. Veroyatnost. Matem. Statist. 60 (2000), 46-58; English transl. in Theor. Probab. Math. Stat. 60 (2001), 53-65. MR 1826141
  • 16. M. V. Kartashov and O. M. Stroyev, The Lundberg approximation for the risk function in an almost homogeneous environment, Teor. Veroyatnost. Matem. Statist. 73 (2005), 63-71; English transl. in Theor. Probab. Math. Stat. 73 (2006), 71-79. MR 2213842 (2007b:62121)
  • 17. N. V. Kartashov, Uniform limit theorems for ergodic stochastic processes and their application in the queueing theory, Section 1.4, Doctoral dissertation, Kiev University, Kiev, 1985. (Russian)
  • 18. N. V. Kartashov, A generalization of Stone's representation and necessary conditions for uniform convergence in the renewal theorem, Teor. Veroyatnost. Matem. Statist. 26 (1983), 49-62; English transl. in Theor. Probab. Math. Stat. 26 (1984), 53-67. MR 664903 (83m:60113)
  • 19. N. V. Kartashov, Equivalence of uniform renewal theorems and their criteria, Teor. Veroyatnost. Matem. Statist. 27 (1983), 51-60; English transl. in Theor. Probab. Math. Stat. 27 (1984), 55-64. MR 673349 (83m:60114)
  • 20. W. Feller, An Introduction to Probability Theory and its Applications, vol. 2, John Wiley & Sons, New York, 1966. MR 0210154 (35:1048)
  • 21. J. Grandell, Aspects of Risk Theory, Springer-Verlag, New York, 1991. MR 1084370 (92a:62151)
  • 22. I. I. Gikhman and A. V. Skorokhod, The Theory of Stochastic Processes, vol. 2, Nauka, Moscow, 1973; English transl., Springer-Verlag, Berlin-Heidelberg-New York, 1975. MR 0341540 (49:6288); MR 0375463 (51:11656)
  • 23. A. A. Borovkov, Stochastic Processes in Queueing Theory, Nauka, Moscow, 1972; English transl., Springer-Verlag, New York-Heidelberg-Berlin, 1976. MR 0315800 (47:4349)
  • 24. C. Stone, On absolutely continuous components and renewal theory, Ann. Math. Stat. 37 (1966), 271-275. MR 0196795 (33:4981)
  • 25. D. Szász, Uniformity in Stone's decomposition of the renewal measure, Ann. Probab. 5 (1977), 560-564. MR 0461697 (57:1681)

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Additional Information

M. V. Kartashov
Affiliation: Department of Probability Theory and Mathematical Statistics, Faculty for Mechanics and Mathematics, National Taras Shevchenko University, Academician Glushkov Avenue 6, Kyiv 03127, Ukraine

Keywords: Volterra equation, renewal theorem, risk function, Lundberg index, Poisson process
Received by editor(s): January 15, 2007
Published electronically: August 4, 2009
Article copyright: © Copyright 2009 American Mathematical Society

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