×

A new method for deriving bounds for integrals with respect to measures allowed to vary under conical and integral constraints. (English) Zbl 0633.65150

Authors’ summary: In insurance literature on applied mathematics in actuarial sciences the theory of convex analysis is applied to so-called stop-loss premiums in case only some moments of the claim distribution are known, possibly combined with other conical characteristics of the distribution. In the present contribution a much simpler method is proposed, based on results from the theory of the problem of moments. The resulting algorithm can handle an arbitrary number of moment constraints, thus considerably generalizing results obtained previously.
Reviewer: W.Uhlmann

MSC:

65C99 Probabilistic methods, stochastic differential equations
65R10 Numerical methods for integral transforms
62P05 Applications of statistics to actuarial sciences and financial mathematics
44A60 Moment problems
62E99 Statistical distribution theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] De Vylder, F., Duality theory for bounds on integrals with applications to stop-loss premiums, Scand. Actuar. J., 129-147 (1983) · Zbl 0522.62087
[2] De Vylder, F.; Goovaerts, M. J., Best bounds on the stop-loss premium in case of known range, expectation, variance and mode of the risk, Insurance Math. Econom., 2, 241-249 (1983) · Zbl 0519.62092
[3] Freud, G., Orthogonal Polynomials (1971), Pergamon Press: Pergamon Press New York · Zbl 0226.33014
[4] Gisler, A., Protokoll der Sitzung vom 11. Mai 1982 der Arbeitsgruppe ASTIN der Vereinigung schweizerischer Versicherungsmathematiker (1982)
[5] Goovaerts, M. J.; De Vylder, F.; Haezendonck, J., Insurance Premiums (1984), North-Holland: North-Holland Amsterdam · Zbl 0532.62082
[6] Goovaerts, M. J.; Kaas, R., Applications of the problem of moments to derive bounds on integrals with integral constraints, Insurance Math. Econom., 4, 99-111 (1985) · Zbl 0559.62086
[7] Jansen, K.; Haezendonck, J.; Goovaerts, M. J., Analytical upper bounds on stop-loss premiums in case of known moments up to the fourth order, Insurance Math. Econom., 5, 315-334 (1986) · Zbl 0607.62129
[8] Kaas, R., Bounds and approximations for some risk-theoretical quantities (1987), University of Amsterdam
[9] Kaas, R.; Goovaerts, M. J., Application of the problem of moments to various insurance problems in non-life, (Proc. NATO-ASI 1985 (1986), Reidel: Reidel Dordrecht), Maratea, Italy · Zbl 0605.62124
[10] Kaas, R.; Goovaerts, M. J., Extremal values of stop-loss premiums under moment constraints, Insurance Math. Econom., 5, 279-287 (1986) · Zbl 0609.62134
[11] Karlin, S.; Studden, W., Tschebysheff Systems: With Applications in Analysis and Statistics (1966), Wiley-Interscience: Wiley-Interscience New York · Zbl 0153.38902
[12] Mack, T., Berechnung der maximalen Stop-Loss-prämie, wenn die ersten drei Momente der Schadenverteilung gegeben sind, Mitt. Verein. Schweiz. Versicherungsmath., 39-55 (1985) · Zbl 0574.62099
[13] Possé, G., Sur quelques applications des fractions continues algébriques (1886), St. Petersburg · JFM 18.0161.02
[14] Shohat, J. A.; Tamarkin, J. D., The Problem of Moments, (Mathematical Surveys I (1943), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 0112.06902
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.