Stochastic convexity of the Poisson mixture model (Q5926521)

Last time the Poisson model is usually rejected when applied to fit real data sets from insurance portfolios. In order to get a model that reflects the different underlying risks, it is common to assume that the distribution of the number of claims incurred by each individual conforms to a Poisson distribution whose parameter \(\theta\) varies from one individual to another and moreover, possible values of this parameter are considered to be the realisation of some positive random variable \(\Theta.\) From a mathematical point of view, this comes down to suppose that the annual number of claims \(N_{\Theta},\) caused by a policyholder is distributed according to a mixed Poisson law, that is \[ P\{N_{\Theta}=k\}=\int_0^{\infty}e^{-\theta}\frac {\theta^k}{k!} dP(\Theta<\theta),\quad k=0, 1,\dots\;. \] This paper is devoted to the study of concrete sides of just this compound Poisson mixture model in an actuarial framework. The amount of premium that an insurance company charges is often of the form \(E=E(\varphi(S_{\Theta}))\) for some function \(\varphi\) and \(S_{\Theta}= \sum_1^{N_{\Theta}}X_i,\) where some random claim amount \(X_i\) corresponds to, say, policyholder \(i.\) When \(E\) is hard or impossible to compute explicitly, then it is useful to have some sharp bounds on \(E.\) The main purpose of this paper is to show that when the function \(\varphi\) is \(s\)-convex (the definition and examples are given), then useful bounds can be obtained. But the authors also describe many interesting features of the model and different topics connected with the above mentioned purpose.