The Principle Of Premiums On Cumulative Claims For Dependency Structures

Lundberg proposed the earliest classical risk model-the compound Poisson model in 1903, where the claim number process is a Poisson counting process, and the claim amounts are independent and identically distributed. In order to better simulate the ac-tual situation of the claim to arrive, Andersen generalized the classical compound Poisson model in 1957, and he first proposed the renewal process. Sklar originally proposed the variable dependence relation can be described by the Copula functions in 1959. For the dependence relations of the internal model, the Copula functions can couple the join-t distribution with the marginal distributions. Nelsen introduced copula functions and described in detail on the joint distribution and marginal distributions. Examples were given for the premium pricing applications. Many experts and scholars study on pricing of the total amount of claims based on dependent risk model, but so far nobody has ever been studying the exponential premium principle. Pricing is the core of the insurance. It is very important to research the scientific premium pricing methods and strengthen the application, which has gained more and more attentions of scholars. Choosing appropriate risk pricing model to evade the risks is concerned by the insurance scholars. In the paper, we give the key assumption, namely, we require that, Therefore, the main objective of the paper is considering two kinds of risk model with dependence structure to derive the premium principles on aggregate claims. The paper is organized as follows.In chapter one, we first introduce the classical risk model with dependence structure, the background and content of Copula, and the compound Poisson risk model, under the dependence between inter-claim time and claims, the moments and moment-generating functions; Especially, under the specific dependence structure, namely, the exponential premium principle and Esscher premium principle of the aggregate claims are derived. Besides, we verify the the moment generating function(mgf)of the compound Poisson process with independent claims and inter-arrival times.In chapter two, the preparatory work is the renewal process in the presence of the dependence structure briefly introduced via the conditional pdf under the assumption of the Farlie-Gumbel-Morgenstern(FGM)copula; In addition, we study the moment generating function(mgf)of Z(t) to derive the premium principles when r.v.W is Erlang(2, λ) distributed. when the claims are the density mixture of exponential and Erlang(2), we also acquire the expression of the density function of the claim time and claim. Besides, the exponential premium principle is given. Finally, the author make a summary of this paper and introduces the future work schedules, namely, we should study the exact solution of premium calculation by utilizing MATLAB.