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Orthogonal polynomial expansions to evaluate stop-loss premiums. (English) Zbl 1451.91166

The authors introduce a numerical method to evaluate the survival function of a compound distribution and the stop-loss premiums associated with a non-proportional global reinsurance treaty. Orthogonal polynomials are used to derive an approximation formula to recover an unknown probability measure from the knowledge of its moments.

MSC:

91G05 Actuarial mathematics
60G40 Stopping times; optimal stopping problems; gambling theory
41A10 Approximation by polynomials
44A10 Laplace transform
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References:

[1] European Insurance and Occupational Pensions Authority, European insurance occupational pensions authority quantitative impact studies V: Technical specifications (2010), European Comission: European Comission Brussels
[2] Denuit, Michel; Dhaene, Jan; Goovaert, Marc J.; Kaas, Rob, Actuarial Theory for Dependent Risk: Measures, Orders and Models (2006), John Wiley & Sons
[3] Cai, Jun; Tan, Ken Seng; Weng, Chengguo; Zhang, Yi, Optimal reinsurance under VaR and CTE risk measures, Insurance Math. Econom., 43, 1, 185-196 (2008) · Zbl 1140.91417
[4] Cheung, Ka Chun, Optimal reinsurance revisited: a geometric approach, Astin Bull., 40, 1, 221-239 (2010) · Zbl 1230.91070
[5] Chi, Yichun; Tan, Ken Seng, Optimal reinsurance under VaR and CVaR risk measures: a simplified approach, Astin Bull., 41, 2, 487-509 (2011) · Zbl 1239.91078
[6] Goffard, Pierre-Olivier; Loisel, Stéphane; Pommeret, Denys, Polynomial approximations for bivariate aggregate claims amount probability distributions, Methodol. Comput. Appl. Probab., 19, 1, 151-174 (2015) · Zbl 1380.65023
[7] Jin, Tao; Provost, Serge B.; Ren, Jiandong, Moment-based density approximations for aggregate losses, Scand. Actuar. J., 2016, 3, 216-245 (2016) · Zbl 1401.91150
[8] Bowers, Newton L., Expansion of probability density functions as a sum of gamma densities with applications in risk theory, Trans. Soc. Actuar., 18, 52, 125-137 (1966)
[9] Asmussen, Søren; Goffard, Pierre-Olivier; Laub, Patrick J., Orthonormal polynomial expansions and lognormal sum densities, (Risk and Stochastics: Ragnar Norberg at 70. Mathematical Finance Economics (2018), World Scientific)
[10] Mnatsakanov, Robert M.; Ruymgaart, L. L.; Ruymgaart, Frits H., Nonparametric estimation of ruin probabilities given a random sample of claims, Math. Methods Statist., 17, 1, 35-43 (2008) · Zbl 1282.62080
[11] Abate, Joseph; Whitt, Ward, The Fourier-series method for inverting transforms of probability distributions, Queueing Syst., 10, 1, 5-87 (1992) · Zbl 0749.60013
[12] Rolski, Tomasz; Schmidli, Hanspeter; Schmidt, Volker; Teugels, Jozef L., (Stochastic Processes for Insurance and Finance. Stochastic Processes for Insurance and Finance, Wiley Series in Probability and Statistics, vol. 505 (2009), John Wiley & Sons) · Zbl 1152.60006
[13] Dufresne, Daniel; Garrido, Jose; Morales, Manuel, Fourier inversion formulas in option pricing and insurance, Methodol. Comput. Appl. Probab., 11, 3, 359-383 (2009) · Zbl 1170.91410
[14] Embrechts, Paul; Frei, Marco, Panjer recursion versus FFT for compound distributions, Math. Methods Oper. Res., 69, 3, 497-508 (2009) · Zbl 1205.91081
[15] Goffard, Pierre-Olivier; Loisel, Stéphane; Pommeret, Denys, A polynomial expansion to approximate the ultimate ruin probability in the compound Poisson ruin model, J. Comput. Appl. Math., 296, 499-511 (2016) · Zbl 1355.60117
[16] Pierre-Olivier Goffard, Patrick J. Laub, Online accompaniment for Orthogonal polynomial expansions to evaluate stop-loss premiums, 2019, Available at https://github.com/Pat-Laub/ActuarialOrthogonalPolynomials. · Zbl 1451.91166
[17] Gzyl, Henryk; Tagliani, Aldo, Determination of the distribution of total loss from the fractional moments of its exponential, Appl. Math. Comput., 219, 4, 2124-2133 (2012) · Zbl 1291.91112
[18] Mnatsakanov, Robert M.; Sarkisian, Khachatur, A note on recovering the distributions from exponential moments, Appl. Math. Comput., 219, 16, 8730-8737 (2013) · Zbl 1288.62020
[19] Asmussen, Søren; Albrecher, Hansjörg, (Ruin Probabilities. Ruin Probabilities, Advanced Series on Statistical Science and Applied Probability, vol. 14 (2010), World Scientific) · Zbl 1247.91080
[20] Gzyl, Henryk; Inverardi, Pier Luigi Novi; Tagliani, Aldo, Determination of the probability of ultimate ruin by maximum entropy applied to fractional moments, Insurance Math. Econom., 53, 2, 457-463 (2013) · Zbl 1304.91242
[21] Mnatsakanov, Robert M.; Sarkisian, Khachatur; Hakobyan, A., Approximation of the ruin probability using the scaled Laplace transform inversion, Appl. Math. Comput., 268, 717-727 (2015) · Zbl 1410.62026
[22] Lefèvre, Claude; Picard, Philippe, A new look at the homogeneous risk model, Insurance Math. Econom., 49, 3, 512-519 (2011) · Zbl 1229.91162
[23] Lefèvre, Claude; Trufin, Julien; Zuyderhoff, Pierre, Some comparison results for finite-time ruin probabilities in the classical risk model, Insurance Math. Econom., 77, Supplement C, 143-149 (2017) · Zbl 1397.91289
[24] Schoutens, Wim, Stochastic Processes and Orthogonal Polynomials, vol. 146 (2012), Springer Science & Business Media: Springer Science & Business Media New York
[25] Diaconis, Persi; Zabell, Sandy, Closed form summation for classical distributions: variations on a theme of de Moivre, Statist. Sci., 6, 3, 284-302 (1991) · Zbl 0955.60500
[26] Szökefalvi-Nagy, Béla, Introduction To Real Functions and Orthogonal Expansions (1965), Akadémiai Kiadó · Zbl 0128.05101
[27] Morris, Carl N., Natural exponential families with quadratic variance functions, Ann. Statist., 10, 1, 65-80 (1982) · Zbl 0498.62015
[28] Letac, Gérard; Mora, Marianne, Natural real exponential families with cubic variance functions, Ann. Statist., 18, 1, 1-37 (1990) · Zbl 0714.62010
[29] Provost, Serge B., Moment-based density approximants, Math. J., 9, 4, 727-756 (2005)
[30] Papush, Dmitry E.; Patrik, Gary S.; Podgaits, Felix, Approximations of the aggregate loss distribution, (CAS Forum (2001), Winter), 175-186
[31] Nishii, Ryuei, Orthogonal Functions of Inverse Gaussian Distributions, 243-250 (1996), Springer · Zbl 0921.60014
[32] Hassairi, Abdelhamid; Zarai, Mohammed, Characterization of the cubic exponential families by orthogonality of polynomials, Ann. Probab., 32, 3, 2463-2476 (2004) · Zbl 1056.62015
[33] Nadarajah, Saralees; Chu, Jeffrey; Jiang, Xiao, On moment based density approximations for aggregate losses, J. Comput. Appl. Math., 298, 152-166 (2016) · Zbl 1341.62057
[34] Szegö, Gabor., Gabor Szegö Orthogonal Polynomials, vol. XXIII (1939), American Mathematical Society Colloquium Publications · Zbl 0023.21505
[35] Kang, John Sang Jin; Provost, Serge B.; Ren, Jiandong, Moment-based density approximation techniques as applied to heavy-tailed distributions, Int. J. Stat. Probab., 8, 3 (2019)
[36] Willmot, Gordon E.; Woo, Jae-Kyung, On the class of Erlang mixtures with risk theoretic applications, N. Am. Actuar. J., 11, 2, 99-115 (2007) · Zbl 1480.91253
[37] Lee, Simon C. K.; Sheldon Lin, X., Modeling and evaluating insurance losses via mixtures of Erlang distributions, N. Am. Actuar. J., 14, 1, 107-130 (2010)
[38] Willmot, Gordon E.; Sheldon Lin, X., Risk modelling with the mixed Erlang distribution, Appl. Stoch. Models Bus. Ind., 27, 1, 2-16 (2011)
[39] Panjer, Harry H.; Willmot, Gordon E., Finite sum evaluation of the negative binomial-exponential model, ASTIN Bull., 12, 2, 133-137 (1981)
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