Some Inequalities for Stop-Loss Premiums

H. Bühlmann, B. Gagliardi, H. U. Gerber and E. Straub

ASTIN Bulletin, 1977, vol. 9, issue 1-2, 75-83

Abstract: In this paper any given risk S (a random variable) is assumed to have a (finite or infinite) mean. We enforce this by imposing E[S−] v((1−z)Q)} is not empty.Proof: a) b) Because of a) E[v(S−zQ)] is always finite or equal to + ∞ If v(− ∞) = − ∞ then E[v(S − zQ)] > v((1 − z)Q) is satisfied for sufficiently small Q. The left hand side of the inequality is a nonincreasing continuous function in P (strictly decreasing if z > 0), while the right hand side is a nondecreasing continuous function in Q (strictly increasing if z > 1).If v(− ∞) = c finite then E[v(S − zQ)] > c(otherwise S would need to be equal to − ∞ with probability 1) and again E[v(S − zQ)] > v((1 − z)Q) is satisfied for sufficiently small Q.

Date: 1977
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