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Modern Stochastics: Theory and Applications


Ruin probabilities as functions of the roots of a polynomial
Volume 10, Issue 3 (2023), pp. 247–266
Pub. online: 15 March 2023      Type: Research Article      Open accessOpen Access
Received
18 December 2022
Revised
8 March 2023
Accepted
9 March 2023
Published
15 March 2023

Abstract

A new formula for the ultimate ruin probability in the Cramér–Lundberg risk process is provided when the claims are assumed to follow a finite mixture of m Erlang distributions. Using the theory of recurrence sequences, the method proposed here shifts the problem of finding the ruin probability to the study of an associated characteristic polynomial and its roots. The found formula is given by a finite sum of terms, one for each root of the polynomial, and allows for yet another approximation of the ruin probability. No constraints are assumed on the multiplicity of the roots and that is illustrated via a couple of numerical examples.

References

[1] 
Andersen, E.S.: On the collective theory of risk in case of contagion between claims. Bull. Inst. Math. Appl. 12(2), 275–279 (1957)
[2] 
Asmussen, S., Albrecher, H.: Ruin Probabilities vol. 14. World scientific Singapore (2010). MR2766220. https://doi.org/10.1142/9789814282536
[3] 
Asmussen, S., Binswanger, K.: Simulation of ruin probabilities for subexponential claims. ASTIN Bull. 27(2), 297–318 (1997)
[4] 
Borchers, H.W.: Pracma: Practical Numerical Math Functions. (2021). R package version 2.3.3. https://CRAN.R-project.org/package=pracma
[5] 
Brousseau, A.: Linear Recursion and Fibonacci Sequences. Fibonacci Assoc. (1971)
[6] 
Chen, Y., Yi, C., Xie, X., Hou, M., Cheng, Y.: Solution of ruin probability for continuous time model based on block trigonometric exponential neural network. Symmetry 12(6), 876 (2020)
[7] 
Choi, S.K., Choi, M.H., Lee, H.S., Lee, E.Y.: New approximations of ruin probability in a risk process. Qual. Technol. Quant. Manag. 7(4), 377–383 (2010)
[8] 
Coen, A., Mena, R.H.: Ruin probabilities for bayesian exchangeable claims processes. Journal of Statistical Planning and Inference 166, 102–115 (2015). MR3390137. https://doi.org/10.1016/j.jspi.2015.01.005
[9] 
Constantinescu, C., Kortschak, D., Maume-Deschamps, V.: Ruin probabilities in models with a markov chain dependence structure. Scandinavian Actuarial Journal 2013(6), 453–476 (2013). MR3176014. https://doi.org/10.1080/03461238.2011.627745
[10] 
Cramér, H.: On the mathematical theory of risk. Skandia Jubilee Volume, Stockholm. Reprinted in: Martin-Löf, A. (Ed.), Cramer, H. (1994). Collected Works. Springer (1930)
[11] 
Cramér, H.: Historical review of filip lundberg’s works on risk theory. Scandinavian Actuarial Journal 1969(sup3), 6–12 (1969). MR0347028. https://doi.org/10.1080/03461238.1969.10404602
[12] 
De Vylder, F.: A practical solution to the problem of ultimate ruin probability. Scandinavian Actuarial Journal 1978(2), 114–119 (1978)
[13] 
Delbaen, F., Haezendonck, J.: Classical risk theory in an economic environment. Insur. Math. Econ. 6(2), 85–116 (1987). MR0896414. https://doi.org/10.1016/0167-6687(87)90019-9
[14] 
Dickson, D.C.M.: Introduction to Ruin Theory, 2nd. edn. International Series on Actuarial Science. Cambridge University Press (2016). https://doi.org/10.1017/9781316650776.007
[15] 
Dufresne, F., Gerber, H.U., Shiu, E.S.: Risk theory with the gamma process. ASTIN Bull. 21(2), 177–192 (1991)
[16] 
Embrechts, P., Schmidli, H.: Ruin estimation for a general insurance risk model. Adv. Appl. Probab. 26(2), 404–422 (1994). MR1272719. https://doi.org/10.2307/1427443
[17] 
Gerber, H.U., Yang, H.: Absolute ruin probabilities in a jump diffusion risk model with investment. N. Am. Actuar. J. 11(3), 159–169 (2007). MR2393866. https://doi.org/10.1080/10920277.2007.10597474
[18] 
Goffard, P.-O., Loisel, S., Pommeret, D.: A polynomial expansion to approximate the ultimate ruin probability in the compound poisson ruin model. J. Comput. Appl. Math. 296, 499–511 (2016). MR3430154. https://doi.org/10.1016/j.cam.2015.06.003
[19] 
Gordienko, E., Vazquez-Ortega, P.: Simple continuity inequalities for ruin probability in the classical risk model. ASTIN Bull. 46(3), 801–814 (2016). MR3551965. https://doi.org/10.1017/asb.2016.10
[20] 
Grigutis, A.: Exact expression of ultimate time survival probability in homogeneous discrete-time risk model. AIMS Math. 8(3), 5181–5199 (2023). MR4525843. https://doi.org/10.3934/math.2023260
[21] 
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge university press (2012). MR2978290
[22] 
Kalashnikov, V., Konstantinides, D.: Ruin under interest force and subexponential claims: a simple treatment. Insur. Math. Econ. 27(1), 145–149 (2000). MR1796976. https://doi.org/10.1016/S0167-6687(00)00045-7
[23] 
Klugman, S.A., Panjer, H.H., Willmot, G.E.: Loss Models: from Data to Decisions vol. 715. John Wiley & Sons (2012). MR3222004
[24] 
Landriault, D., Renaud, J.-F., Zhou, X.: An insurance risk model with parisian implementation delays. Methodol. Comput. Appl. Probab. 16(3), 583–607 (2014). MR3239810. https://doi.org/10.1007/s11009-012-9317-4
[25] 
Lee, S.C., Lin, X.S.: Modeling and evaluating insurance losses via mixtures of erlang distributions. N. Am. Actuar. J. 14(1), 107–130 (2010). MR2720423. https://doi.org/10.1080/10920277.2010.10597580
[26] 
Lefèvre, C., Picard, P.: A nonhomogeneous risk model for insurance. Comput. Math. Appl. 51(2), 325–334 (2006). MR2203083. https://doi.org/10.1016/j.camwa.2005.11.005
[27] 
Li, J., Liu, Z., Tang, Q.: On the ruin probabilities of a bidimensional perturbed risk model. Insur. Math. Econ. 41(1), 185–195 (2007). MR2324573. https://doi.org/10.1016/j.insmatheco.2006.10.012
[28] 
Lundberg, F.: Approximerad framställning afsannollikhetsfunktionen: Ii. återförsäkring af kollektivrisker.... PhD thesis, Almqvist & Wiksells Boktr. (1903)
[29] 
Lundberg, F.: Försäkringsteknisk Riskutjämning. F. Englunds boktryckeri (1926)
[30] 
Miljkovic, T., Grün, B.: Modeling loss data using mixtures of distributions. Insur. Math. Econ. 70, 387–396 (2016). MR3543061. https://doi.org/10.1016/j.insmatheco.2016.06.019
[31] 
Mnatsakanov, R.M., Sarkisian, K., Hakobyan, A.: Approximation of the ruin probability using the scaled laplace transform inversion. Appl. Math. Comput. 268, 717–727 (2015). MR3399457. https://doi.org/10.1016/j.amc.2015.06.087
[32] 
Peralta, O., Rojas-Nandayapa, L., Xie, W., Yao, H.: Approximation of ruin probabilities via erlangized scale mixtures. Insur. Math. Econ. 78, 136–156 (2018). MR3761073. https://doi.org/10.1016/j.insmatheco.2017.12.005
[33] 
Philipson, C.: A note on moments of a poisson probability distribution. Scandinavian Actuarial Journal 1963(3-4), 243–244 (1963). MR0174073. https://doi.org/10.1080/03461238.1963.10410613
[34] 
R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2020). R Foundation for Statistical Computing. https://www.R-project.org/
[35] 
Rincón, L., Santana, D.J.: Ruin probability for finite erlang mixture claims via recurrence sequences. Methodol. Comput. Appl. Probab., 24, 2213–2236 (2022). MR4457584. https://doi.org/10.1007/s11009-021-09913-2
[36] 
Rincón, L., Santana, D.J.: Ruin probability for finite negative binomial mixture claims via recurrence sequences. Commun. Stat., Theory Methods, 1–17 (2022). https://doi.org/10.1080/03610926.2022.2087091
[37] 
Santana, D.J., González-Hernández, J., Rincón, L.: Approximation of the ultimate ruin probability in the classical risk model using erlang mixtures. Methodol. Comput. Appl. Probab. 19(3), 775–798 (2017). MR3683971. https://doi.org/10.1007/s11009-016-9515-6
[38] 
Schassberger, R.: Warteschlangen. Springer (1973). MR0365768
[39] 
Sedgewick, R., Flajolet, P.: An Introduction to the Analysis of Algorithms. Pearson Education India (2013). MR2876111. https://doi.org/10.1051/ita/2011110
[40] 
Tamturk, M., Utev, S.: Ruin probability via quantum mechanics approach. Insur. Math. Econ. 79, 69–74 (2018). MR3771914. https://doi.org/10.1016/j.insmatheco.2017.12.009
[41] 
Thorin, O.: Some remarks on the ruin problem in case the epochs of claims form a renewal process. Scandinavian Actuarial Journal 1970(1-2), 29–50 (1970). MR0290480. https://doi.org/10.1080/03461238.1970.10405645
[42] 
Tijms, H.C.: Stochastic Modelling and Analysis: a Computational Approach. John Wiley & Sons, Inc. (1986). MR0847718
[43] 
Verbelen, R., Gong, L., Antonio, K., Badescu, A., Lin, S.: Fitting mixtures of erlangs to censored and truncated data using the em algorithm. ASTIN Bull. 45(3), 729–758 (2015). MR3394072. https://doi.org/10.1017/asb.2015.15
[44] 
Willmot, G., Lin, S.: Risk modelling with the mixed erlang distribution. Appl. Stoch. Models Bus. Ind. 27(1), 2–16 (2011). MR2752449. https://doi.org/10.1002/asmb.838
[45] 
Willmot, G., Woo, J.K.: On the class of erlang mixtures with risk theoretic applications. N. Am. Actuar. J. 11(2), 99–115 (2007). MR2380721. https://doi.org/10.1080/10920277.2007.10597450
[46] 
You, H., Guo, J., Jiang, J.: Interval estimation of the ruin probability in the classical compound Poisson risk model. Comput. Stat. Data Anal. 144, 106890 (2020). MR4040161. https://doi.org/10.1016/j.csda.2019.106890

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© 2023 The Author(s). Published by VTeX
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Keywords
Ruin probability Cramér–Lundberg risk model Erlang mixture distribution recurrence sequences 91B05 91G05 11B37

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The authors received no financial support for the research, authorship, and/or publication of this article.

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