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Application of the problem of moments to derive bounds on integrals with integral constraints. (English) Zbl 0559.62086

Recently a lot of results have been obtained for bounds on stop-loss premiums in case of incomplete information on the claim distribution. As a consequence some extremal distributions (depending on the retention limit) have been characterized. The extremal distributions for the stop- loss ordering in case of fixed values of the retention limit are obtained by means of deep results from the theory of convex analysis.
In the present contribution it is shown, by means of some results from the problem of moments, how bounds on integrals with integral constraints can be obtained. We assume only the knowledge of the moments \(\mu_ 0,\mu_ 1,...,\mu_ n\).

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
26A99 Functions of one variable
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
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References:

[1] Freud, G., Orthogonal Polynomials (1971), Pergamon Press: Pergamon Press New York · Zbl 0226.33014
[2] Goovaerts, M. J.; Haezendonck, J.; De Vylder, F., Numerical best bounds on stop-loss premiums, Insurance Mathematics and Economics, 1, 287-302 (1982) · Zbl 0498.62089
[3] Goovaerts, M. J.; Haezendonck, J.; de Vylder, F., Insurance Premiums (1983), North-Holland: North-Holland Amsterdam · Zbl 0532.62082
[4] Pólya, G.; Szegö, G., (Aufgaben und Lehrsätze aus der Analysis, Vol. II (1925), Springer: Springer Berlin) · JFM 51.0173.01
[5] Shohat, J. A.; Tamarkin, J. D., The Problem of Moments, (Mathematical Surveys I (1963), American Mathematical Society) · Zbl 0112.06902
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