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Asymptotic results for renewal risk models with risky investments

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  • Albrecher, Hansjoerg
  • Constantinescu, Corina
  • Thomann, Enrique

Abstract

We consider a renewal jump–diffusion process, more specifically a renewal insurance risk model with investments in a stock whose price is modeled by a geometric Brownian motion. Using Laplace transforms and regular variation theory, we introduce a transparent and unifying analytic method for investigating the asymptotic behavior of ruin probabilities and related quantities, in models with light- or heavy-tailed jumps, whenever the distribution of the time between jumps has rational Laplace transform.

Suggested Citation

  • Albrecher, Hansjoerg & Constantinescu, Corina & Thomann, Enrique, 2012. "Asymptotic results for renewal risk models with risky investments," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3767-3789.
    • Handle: RePEc:eee:spapps:v:122:y:2012:i:11:p:3767-3789
    DOI: 10.1016/j.spa.2012.05.017
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    References listed on IDEAS

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    1. Anna Frolova & Serguei Pergamenshchikov & Yuri Kabanov, 2002. "In the insurance business risky investments are dangerous," Finance and Stochastics, Springer, vol. 6(2), pages 227-235.
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    7. Kostadinova, Radostina, 2007. "Optimal investment for insurers when the stock price follows an exponential Lévy process," Insurance: Mathematics and Economics, Elsevier, vol. 41(2), pages 250-263, September.
    8. Jostein Paulsen, 2008. "Ruin models with investment income," Papers 0806.4125, arXiv.org, revised Dec 2008.
    9. Albrecher, Hansjörg & Constantinescu, Corina & Pirsic, Gottlieb & Regensburger, Georg & Rosenkranz, Markus, 2010. "An algebraic operator approach to the analysis of Gerber-Shiu functions," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 42-51, February.
    10. Tang, Qihe & Wei, Li, 2010. "Asymptotic aspects of the Gerber-Shiu function in the renewal risk model using Wiener-Hopf factorization and convolution equivalence," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 19-31, February.
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    12. Grandits, Peter, 2004. "A Karamata-type theorem and ruin probabilities for an insurer investing proportionally in the stock market," Insurance: Mathematics and Economics, Elsevier, vol. 34(2), pages 297-305, April.
    13. Tang, Qihe & Tsitsiashvili, Gurami, 2003. "Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks," Stochastic Processes and their Applications, Elsevier, vol. 108(2), pages 299-325, December.
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    Cited by:

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    2. Kohatsu-Higa, Arturo & Nualart, Eulalia & Tran, Ngoc Khue, 2022. "Density estimates for jump diffusion processes," Applied Mathematics and Computation, Elsevier, vol. 420(C).
    3. Jing Wang & Zbigniew Palmowski & Corina Constantinescu, 2021. "How Much We Gain by Surplus-Dependent Premiums—Asymptotic Analysis of Ruin Probability," Risks, MDPI, vol. 9(9), pages 1-17, August.
    4. Hansjörg Albrecher & Eleni Vatamidou, 2019. "Ruin Probability Approximations in Sparre Andersen Models with Completely Monotone Claims," Risks, MDPI, vol. 7(4), pages 1-14, October.
    5. Eberlein, Ernst & Kabanov, Yuri & Schmidt, Thorsten, 2022. "Ruin probabilities for a Sparre Andersen model with investments," Stochastic Processes and their Applications, Elsevier, vol. 144(C), pages 72-84.

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