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A Novel Approximation Method for Computing the Adjustment Coefficient of a Nonlinear Cramér-Lundberg Risk Model with Gamma Claims

Author

Listed:
  • Basak Gever Ekinci

    (University of Turkish Aeronautical Association)

  • Zulfiye Hanalioglu

    (Karabuk University)

  • Tahir Khaniyev

    (TOBB University of Economics and Technology
    Azerbaijan State University of Economics)

Abstract

This study considers a non-linear Cramér-Lundberg risk model and examines the adjustment coefficient $$\varvec{(r)}$$ ( r ) when the claims have gamma distribution. The linear models are not always adequate because an insurance company’s premium income does not always increase linearly. Therefore, in this study, a more realistic non-linear Cramér-Lundberg risk model is mathematically constructed. Then, the ruin probability of this non-linear risk model is studied when the premium function is in the form of square root function, i.e., $$\varvec{p}\varvec{(t)}\varvec{=}\varvec{c}\varvec{\sqrt{t}}$$ p ( t ) = c t . It leads to analyzing the adjustment coefficient $$\varvec{(r)}$$ ( r ) , as examining this coefficient is required for finding an upper bound while investigating the ruin probability. However, in general case, it is a challenging procedure to calculate the exact value of $$\varvec{r}$$ r from an integral equation. Thus, in this study, the adjustment coefficient $$\varvec{r}$$ r is explored by computational methods and a new approximate formula for the practical calculation of the adjustment coefficient is proposed. Moreover, an implementation of the obtained approximate formula, which investigates ruin probability, is included as an example at the end of the paper.

Suggested Citation

  • Basak Gever Ekinci & Zulfiye Hanalioglu & Tahir Khaniyev, 2025. "A Novel Approximation Method for Computing the Adjustment Coefficient of a Nonlinear Cramér-Lundberg Risk Model with Gamma Claims," Methodology and Computing in Applied Probability, Springer, vol. 27(3), pages 1-19, September.
    • Handle: RePEc:spr:metcap:v:27:y:2025:i:3:d:10.1007_s11009-025-10194-2
    DOI: 10.1007/s11009-025-10194-2
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    References listed on IDEAS

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    1. Malinovskii, Vsevolod K., 2014. "Improved asymptotic upper bounds on the ruin capital in the Lundberg model of risk," Insurance: Mathematics and Economics, Elsevier, vol. 55(C), pages 301-309.
    2. Gauchon, Romain & Loisel, Stéphane & Rullière, Jean-Louis & Trufin, Julien, 2020. "Optimal prevention strategies in the classical risk model," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 202-208.
    3. Hailiang Yang, 1999. "Non-exponential Bounds for Ruin Probability with Interest Effect Included," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 1999(1), pages 66-79.
    4. Anastasiadis, Simon & Chukova, Stefanka, 2012. "Multivariate insurance models: An overview," Insurance: Mathematics and Economics, Elsevier, vol. 51(1), pages 222-227.
    5. Chadjiconstantinidis, Stathis & Politis, Konstadinos, 2007. "Two-sided bounds for the distribution of the deficit at ruin in the renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 41(1), pages 41-52, July.
    6. Rob Kaas & Marc Goovaerts & Jan Dhaene & Michel Denuit, 2008. "Modern Actuarial Risk Theory," Springer Books, Springer, edition 2, number 978-3-540-70998-5, March.
    7. Linlin Tian & Lihua Bai, 2022. "Minimizing ruin probability under the Sparre Anderson model," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 51(6), pages 1622-1636, March.
    8. Martire, Antonio Luciano, 2022. "Volterra integral equations: An approach based on Lipschitz-continuity," Applied Mathematics and Computation, Elsevier, vol. 435(C).
    9. Cohen, Asaf & Young, Virginia R., 2020. "Rate of convergence of the probability of ruin in the Cramér–Lundberg model to its diffusion approximation," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 333-340.
    10. Corina Constantinescu & Gennady Samorodnitsky & Wei Zhu, 2018. "Ruin probabilities in classical risk models with gamma claims," Scandinavian Actuarial Journal, Taylor & Francis Journals, vol. 2018(7), pages 555-575, August.
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