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Second Order Asymptotics for Infinite-Time Ruin Probability in a Compound Renewal Risk Model

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Abstract

Consider a compound renewal risk model, in which a single accident may cause more than one claim. Under the condition that the common distribution of the individual claims is second order subexponential, we establish a second order asymptotic formula for the infinite-time ruin probability. Compared with the traditional ones, our second order asymptotic result is more precise and effective, which can be demonstrated by the numerical studies.

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References

  • Aleškevičienė A, Leipus R, Šiaulys J (2008) Tail behavior of random sums under consistent variation with applications to the compound renewal risk model. Extremes 11:261–279

    Article  MathSciNet  Google Scholar 

  • Asmussen S, Foss S, Korshunov D (2003) Asymptotics for sums of random variables with local subexponential behaviour. J Theoret Probab 16:489–518

    Article  MathSciNet  Google Scholar 

  • Asmussen S, Albrecher H (2010) Ruin probabilities. World Scientific Publishing, Singapore

    Book  Google Scholar 

  • Baltrūnas A, Daley DJ, Klüppelberg C (2004) Tail behaviour of the busy period of a GI/GI/1 queue with subexponential service times. Stochas Process Appl 111(2):237–258

    Article  MathSciNet  Google Scholar 

  • Baltrūnas A (2005) Second order behaviour of ruin probabilities in the case of large claims. Insur Math Econ 36(3):485–498

    Article  MathSciNet  Google Scholar 

  • Cang Y, Yang Y, Shi X (2020) A note on the uniform asymptotic behavior of finite-time ruin probability in a nonstandard renewal risk model. Lithuanion Math J 60:161–172

    Article  MathSciNet  Google Scholar 

  • Chistyakov VP (1964) A Theorem on sums of independent positive random variables and its applications to branching random processes. Theory Probab Appl 9(4):640–648

    Article  MathSciNet  Google Scholar 

  • Embrechts P, Klüppelberg C, Mikosch T (1997) Modelling extremal events for insurance and finance. Springer, Berlin

    Book  Google Scholar 

  • Embrechts P, Veraverbeke N (1982) Estimates for probability of ruin with special emphasis on the possibility of large claims. Insur Math Econ 1(1):55–72

    Article  MathSciNet  Google Scholar 

  • Foss S, Korshunov D, Zachary S (2011) An introduction to heavy-tailed and subexponential distributions. Springer, New York

    Book  Google Scholar 

  • Kass R, Tang Q (2005) A large deviation result for aggregate claims with dependent claim occurrences. Insur Math Econ 36(3):251–259

    Article  MathSciNet  Google Scholar 

  • Kočetova J, Leipus R, Šiaulys J (2009) A property of the renewal counting process with application to the finite-time ruin probability. Lithuanian Math J 49:55–61

    Article  MathSciNet  Google Scholar 

  • Korshunov D (2018) On subexonential tails for the maxima of negatively deriven compound renewmal and lévy processes. Stoch Process Appl 128 (4):1316–1332

    Article  Google Scholar 

  • Kortschak D, Hashorva E (2014) Second order asymptotics of aggregated log-elliptical risk. Method Comput Appl Probab 16:969–985

    Article  MathSciNet  Google Scholar 

  • Leipus R, Šiaulys J (2007) Asymptotic behaviour of the finite-time ruin probability under subexponential claim sizes. Insur Math Econ 40(3):498–508

    Article  MathSciNet  Google Scholar 

  • Lin J (2012a) Second order asymptotics for ruin probabilities in a renewal risk model with heavy-tailed claims. Insur Math Econ 51(2):422–429

    Article  MathSciNet  Google Scholar 

  • Lin J (2012b) Second order subexponential distributions with finite mean and their applications to subordinated distributions. J Theoret Probab 25:834–853

    Article  MathSciNet  Google Scholar 

  • Lin J (2014) Second order tail behaviour for heavy-tailed sums and their maxima with applications to ruin theory. Extremes 17:247–262

    Article  MathSciNet  Google Scholar 

  • Lin J (2019) Second order tail approximation for the maxima of randomly weighted sums with applications to ruin theory and numerical examples. Stat Probab Lett 153:37–47

    Article  MathSciNet  Google Scholar 

  • Lin J (2021) Second order asymptotics for ruin probabilities of the delayed renewal risk model with heavy-tailed claims. Commun Stat Theory Methods 50 (5):1200–1209

    Article  MathSciNet  Google Scholar 

  • Tang Q, Su C, Jiang T, Zhang J (2001) Large deviations for heavy-tailed random sums in compound renewal model. Stat Probab Lett 52:91–100

    Article  MathSciNet  Google Scholar 

  • Wang Y, Cui Z, Wang K, Ma X (2012) Uniform asymptotics of the finite-time ruin probability for all times. J Math Anal Appl 390:208–223

    Article  MathSciNet  Google Scholar 

  • Yang Y, Leipus R, Šiaulys J, Cang Y (2011) Uniform estimates for the finite-time ruin probability in the dependent renewal risk model. J Math Anal Appl 383:215–225

    Article  MathSciNet  Google Scholar 

  • Yang Y, Lin J, Huang C, Ma X (2012) The finite-time ruin probability in two non-standard renewal risk models with constant interest rate and dependent subexponential claims. J Korean Stat Soc 41:213–224

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors are most grateful to the referee for his/her very thorough reading of the paper and valuable suggestions, which greatly improve the original results and presentation of this paper. This paper was supported by the Humanities and Social Sciences Foundation of the Ministry of Education of China (No. 20YJA910006), Natural Science Foundation of Jiangsu Province of China (No. BK20201396), Natural Science Foundation of the Jiangsu Higher Education Institutions (No. 19KJA180003), Postgraduate Education Reform Project of Jiangsu Province (No. JGLX19_091), the Project of Construction for Superior Subjects of Mathematics/Statistics of Jiangsu Higher Education Institutions.

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  1. School of Statistics and Mathematics, Nanjing Audit University, Nanjing, 211815, People’s Republic of China

    Yang Yang, Xinzhi Wang & Shaoying Chen

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  1. Yang Yang
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  2. Xinzhi Wang
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  3. Shaoying Chen
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Correspondence to Yang Yang.

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Yang, Y., Wang, X. & Chen, S. Second Order Asymptotics for Infinite-Time Ruin Probability in a Compound Renewal Risk Model. Methodol Comput Appl Probab 24, 1221–1236 (2022). https://doi.org/10.1007/s11009-021-09862-w

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  • DOI: https://doi.org/10.1007/s11009-021-09862-w

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