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Ultimate Ruin Probability for Benktander Gibrat Risk Model

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Applied Statistical Methods (ISGES 2020)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 380))

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Abstract

In actuarial science and finance, the derivation of ultimate ruin probability for various loss distributions is of key interest. There are many methods available in literature for evaluating ultimate ruin probability for different distributions. Probability of ultimate ruin is derived for a risk model under Benktander Gibrat (BG) distribution, also known as Benktander Type I distribution. Laplace transform, generalized exponential integrals, MeijerG function and Bromwich Integral are used to find ultimate ruin probability.

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References

  • Abramowitz, M., & Stegun, I. A. (1964). Handbook of mathematical functions. New York: Dover Publications.

    Google Scholar 

  • Adekambi, F., & Essiomle, K. (2020). Ruin probability for stochastic flows of financial contract under phase-type distribution. Risks, 8(53), 1–21.

    Google Scholar 

  • Arnold, B. C. (2015). Pareto distributions (2nd ed.). Florida: CRC Press.

    Google Scholar 

  • Asmussen, S., & Albrecher, H. (2010). Ruin probabilities (2nd ed., Vol. 14). Advanced Series on Statistical Science and Applied Probability. Singapore: World Scientific.

    Google Scholar 

  • Baltrünas, A. (1999). Second order behaviour of ruin probabilities. Scandinavian Actuarial Journal, 2, 120–133.

    Google Scholar 

  • Barbe, P., & McCormick, W. P. (2009). Asymptotic expansions for infinite weighted convolutions of heavy tail distributions and applications (Vol. 197). USA: American Mathematical Society.

    Google Scholar 

  • Bateman, H., & Erdély, A. (1953). Higher transcendental functions (Vol. I). New York: Mc-Graw Hill.

    Google Scholar 

  • Beirlant, J., Teugels, J., & Vynckier, P. (1996). Practical analysis of extreme values. Leuven: Leuven University Press.

    Google Scholar 

  • Benktander, G. (1970): Schadenverteilungen nach Grösse in der Nicht-Lebensversicherung [Loss Distributions by Size in Non-life Insurance], Bulletin of the Swiss Association of Actuaries (in German), 263–283.

    Google Scholar 

  • Benktander, G., & Segerdahl, C. O. (1960): On the analytical representation of claim distributions with special reference to excess of loss reinsurance. In: Transaction of the 16th International Congress of Actuaries, vol. 1, pp. 626–648.

    Google Scholar 

  • Bohman, H. (1971). Ruin probabilities. Scandinavian Actuarial Journal, 54, 159–163.

    Article  Google Scholar 

  • Bohman, H. (1975). Numerical inversion of characteristic functions. Scandinavian Actuarial Journal, 2, 121–124.

    Article  MathSciNet  Google Scholar 

  • Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A., & Nesbitt, C. J. (1997). Actuarial mathematics. USA: The Society of Actuaries.

    Google Scholar 

  • Brown, J. W., & Churchill, R. V. (2013). Complex variables and applications (9th ed.). New York: McGraw-Hill.

    Google Scholar 

  • Davies, B. J. (2002), Integral transforms and their applications (3rd ed.). Berlin, New York: Springer-Verlag.

    Google Scholar 

  • Daykin, C. D., Pentikäinen, T., & Pesonen, M. (1994). Practical risk theory for actuaries. London: Chapman and Hall.

    Google Scholar 

  • Dickson, D. C. M. (1989): Recursive calculation of the probability and severity of ruin. Insurance: Mathematics and Economics, 8, 145–148.

    Google Scholar 

  • Dickson, D. C. M., & Waters, H. R. (1991). Recursive calculation of survival probabilities. ASTIN Bulletin, 21, 199–221.

    Article  Google Scholar 

  • Dickson, D. C. M., Egidio dos Reis, A. D., & Waters, H. R. (1995). Some stable algorithms in ruin theory and their application. ASTIN Bulletin, 25, 153–175.

    Article  Google Scholar 

  • Dufresne, F., & Gerber, H. U. (1988). The probability and severity of ruin for combination of exponential claim amount distributions and their translations. Insurance: Mathematics and Economics 7, 75-80.

    Google Scholar 

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

    Google Scholar 

  • Embrechets, P., & Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the probability of large claims. Insurance: Mathematics and Economics, 1(1), 55–72.

    Google Scholar 

  • Gerber, H. U. (1979). An introduction to mathematical risk theory. Huebner Foundation Monographs. Philadelphia, S.S: University of Pennsylvania.

    Google Scholar 

  • Goovaerts, M. J., & De Vijylder, F. (1984). A stable recursive algorithm for evaluation of ultimate ruin probabilities. ASTIN Bulletin, 14, 53–59.

    Article  Google Scholar 

  • Grandell, J., & Segerdahl, C. O. (1971). A comparison of some applications of ruin probabilities. Scandinavian Actuarial Journal, 3–4, 143–158.

    Article  Google Scholar 

  • Grübel, R. (1987). On subordinated distributions and generalized renewal measures. The Annals of Probability, 15(1), 394–415.

    Article  MathSciNet  Google Scholar 

  • Hazewinkel, M. (ed.) (2001). Volterra equation,Encyclopedia of Mathematics, Springer Science+Business Media B.V./Kluwer Academic Publishers.

    Google Scholar 

  • Hogg, R. V., & Klugman, S. A. (1984). Loss distributions. New York: John Wiley & Sons.

    Google Scholar 

  • Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Distribution in statistics: continuous univariate distributions (Vol. 2). John Wiley, New York: Second Edition.

    Google Scholar 

  • Lambert, P. J. (1993). The distribution and redistribution of income (2nd ed.). Manchester: Manchester University Press.

    Google Scholar 

  • Kleiber, C., & Kotz, S. (2003). Statistical size distributions in economics and actuarial sciences. New Jersey: Hoboken.

    Google Scholar 

  • Klugman, S. A., Panjer, H. H., & Willmot, G. E. (1998). Loss models. New York: Wiley.

    Google Scholar 

  • Panjer, H. H. (1986). Direct calculation of ruin probabilities. Journal of Risk and Insurance, 53, 521–529.

    Article  Google Scholar 

  • Panjer, H. H., & Wang, S. (1993). On the stability of recursive formulas. ASTIN Bulletin, 23, 227–258.

    Article  Google Scholar 

  • Ramsay, C. M., & Usábel, M. (1997). Calculating ruin probabilities via product integration. ASTIN Bulletin, 27(2), 263–271.

    Article  Google Scholar 

  • Ramsay, C. M. (2003). A solution to the ruin problem for pareto distributions. Insurance: Mathematics and Economics, 33(1), 109–116.

    Google Scholar 

  • Ramsay, C. M. (2009). The distribution of compound sums of pareto distributed losses. Scandinavian Actuarial Journal, 1, 27–37.

    Article  MathSciNet  Google Scholar 

  • Schiff, J. L. (1999). The laplace transform. New York: Springer.

    Google Scholar 

  • Seal, H. L. (1969). Stochastic theory of a risk business. New York: Wiley.

    Google Scholar 

  • Seal, H. L. (1977). Numerical inversion of characteristic functions. Scandinavian Actuarial Journal, 1, 48–53.

    Article  Google Scholar 

  • Thorin, O., & Wikstad, N. (1977). Calculation of ruin probabilities when the claim distribution is lognormal. ASTIN Bulletin, 9, 231–246.

    Article  Google Scholar 

  • Usábel, M. (1999). Calculating multivariate ruin probabilities via gaver-stehfest inversion technique. Insurance: Mathematics and Economics, 25, 133–142.

    Google Scholar 

  • Usábel, M. (2001). Ultimate ruin probabilities for generalized gamma-convolution claim sizes. ASTIN Bulletin, 31(1), 59–79.

    Article  MathSciNet  Google Scholar 

  • Von Bahr, B. (1975). Asymptotic ruin probabilities when exponential moments do not exist. Scandinavian Actuarial Journal, 6–10.

    Google Scholar 

  • Wei, L., & Hai-liang, Y. (2004). Explicit expressions for the ruin probabilities of Erlang risk processes with Pareto individual claim distributions. Acta Mathematicae Applicatae Sinica, 20, 495–506.

    Article  MathSciNet  Google Scholar 

  • Willmot, G. E. (1988): Further use of shiu’s approach to the evaluation of ultimate ruin probabilities. Insurance: Mathematics and Economics, 7, 275–281.

    Google Scholar 

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Acknowledgements

The authors are grateful to the Editor and refrees for their valuable comments and suggestions which have helped in improving the contents of this chapter.

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Authors and Affiliations

  1. Department of Statistics, Panjab University, Chandigarh, 160014, India

    Kanchan Jain

  2. Department of Mathematics and Statistics, Central University of Punjab, Bathinda, 151001, India

    Harmanpreet Singh Kapoor

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  1. Kanchan Jain
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  2. Harmanpreet Singh Kapoor
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Editor information

Editors and Affiliations

  1. Pune, Maharashtra, India

    David D. Hanagal

  2. Department of Statistics, Savitribai Phule Pune University, Pune, Maharashtra, India

    Raosaheb V. Latpate

  3. Division of Forestry Statistics, Indian Council of Forestry Research and Education, Dehradun, Uttarakhand, India

    Girish Chandra

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Jain, K., Kapoor, H.S. (2022). Ultimate Ruin Probability for Benktander Gibrat Risk Model. In: Hanagal, D.D., Latpate, R.V., Chandra, G. (eds) Applied Statistical Methods. ISGES 2020. Springer Proceedings in Mathematics & Statistics, vol 380. Springer, Singapore. https://doi.org/10.1007/978-981-16-7932-2_11

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