A Note on the Shape of the Probability Weighting Function

The focus of this contribution is on the transformation of objective probability, which in Prospect Theory is commonly referred as probability weighting. Empirical evidence suggests a typical inverse-S shaped function: decision makers tend to overweight small probabilities, and underweight medium and high probabilities; moreover, the probability weighting function is initially concave and then convex. We apply different parametric weighting functions proposed in the literature to the evaluation of derivative contracts and to insurance premium principles

Optimal Reinsurance with One Insurer and Multiple Reinsurers

In this paper, we consider a one-period optimal reinsurance design model with n reinsurers and an insurer. For very general preferences of the insurer, we obtain that there exists a very intuitive pricing formula for all reinsurers that use a distortion premium principle. The insurer determines its optimal risk that it wants to reinsure via this pricing formula. This risk it wants to reinsure is then shared by the reinsurers via tranching. The optimal ceded loss functions among multiple reinsurers are derived explicitly under the additional assumptions that the insurer’s preferences are given by an inverse-S shaped distortion risk measure and that the reinsurer’s premium principles are some functions of the Conditional Value-at-Risk. We also demonstrate that under some prescribed conditions, it is never optimal for the insurer to cede its risk to more than two reinsurers

On Fatou-type lemma for monotone moments of weakly convergent random variables

Sufficient conditions for convergence of monotone moments of weakly convergent random variables, concerning the rate of convergence, are given. They are often more convenient than the necessary and sufficient uniform integrability condition. Some asymptotic evaluations for inverse moments are presented.Weak convergence Fatous lemma Inverse moments Cramers theorem Berry-Esséen's bound

An extension of the Erdös-Neveu-Rényi theorem with applications to order statistics

The necessary and sufficient conditions are given for some stochastic process to be an empirical distribution function from some exchangeable random variables. The result is applied to establish sharp lower and upper bounds for order statistics based on possibly dependent random variables.Dependent random variables Exchangeable random variables Empirical distribution function Order statistics Bonferroni-type inequalities

A note on weighted premium calculation principles

A prominent problem in actuarial science is to determine premium calculation principles that satisfy certain criteria. Goovaerts et al. [Goovaerts, M. J., De Vylder, F., Haezendonck, J., 1984. Insurance Premiums: Theory and Applications. North-Holland, Amsterdam, p. 84] establish an optimality-type characterization of the Esscher premium principle, but unfortunately their result is not true. In this note we propose a modified statement of this result

Sharp exponential and entropy bounds on expectations of generalized order statistics

generalized order statistics, order statistics, records, entropy, Moriguti's inequality, Young's inequality, Gumbel distribution, Pareto distribution,

The law of the iterated logarithm for Lp-norms of empirical processes

Let X1,X2,... be a sequence of i.i.d. random variables with a common continuous distribution function F and let p [greater-or-equal, slanted] 1. It is shown that where Fn denotes the empirical distribution function of the sample X1,X2,...,Xn60F25, 62G30 Empirical distribution function Law of iterated logarithm (LIL) Lp-limit theorems

Bounds for expectations of concomitants

Order statistics, Concomitants, Copulas, Finite populations,