On Optimal Insurance Decision With Dependent Risks

The independence assumption for claim risks is of importance in traditional insurance risk theory,and is also the foundation of risk estimating and decision making of an insurance company.This assumption,however,is only to facilitate the mathematical processing.In the insurance practice,the claim amount in different time,the claim numbers of different insurance businesses and the relationship between claim amount and the time of arrival interval often present dependence.On the other hand,the assumption of identically distribution for claim amount in the classical risk model generally only applies to homogeneous risk,while the reality of modern diversified insurance company is planted.Therefore,adopting dependent risk model(multiple businesses)to characterize insurance risks,and basing on this to discuss the optimal decision for insurance companys have very important theoretical and practical significance.The problem of optimal insurance decision mainly involves three aspects consisting of optimal reinsurance,optimal investment and optimal dividend.Considering that the risk models should be as close to the actual as possilbe,this thesis studys the optimal strategies of the three kinds of economic business with dependent claim risks,respectively,and analysis the dynamic effect of dependence.We expected the obtained result can provide the insurance company some theoretical basis and instructive reference to realize the optimal decision-making.The main work and achievements are as follows:1.Study on optimal reinsurance with dependent risks under a unified optimization criterion.First of all,the traditional binary compound Poisson model is improved,the claim amounts and claim numbers are both supposed to be positively dependent through the stochastic ordering,the optimal reinsurance strategy of the improved dependent bivariate-insurance model is studied.The optimality of excess of loss reinsurance form is proved,and the explicit expression of the optimal retention vector is obtained under the optimization criterion of minimizing the variance and maximizing the parabolic expected utility.Then,aiming at dependent n(?2)businesses risk model to study the determination of optimal retention vector,the results are presented with equations.Meanwhile,the explicit expressions are obtained in two-dimensional case,for high-dimensional case,equations easier to solve are given respectively under the particular criteria of minimizing the variance and maximizing the expected exponential utility,of retained risks in a particular dependence structure,and the numerical solutions are given by intelligent algorithm combing with numerical examples.Finally,considering the influence of stochastic catastrophe,an adjustable reinsurance model is established under the conditions that the claim risks are time-varying and dependent.Through analysis,it is found that the unified objective functions under arbitrary dependence structure are decreasing first and then turn out to be increasing as the retention level of catastrophe risk increases,thus the explicit expression of the retention vector is given in a general form.Example Comparing reflects the superiority of dynamic reinsurance.2.Study on the optimal insurance investment with dependent risks under tail risk constraint.Adopting tail risk measure and policy supervision to constraint integrated risk,we study the optimal investment strategies which maximize the expected terminal wealth with dependent claims,dependent claim numbers and dependence between claim amount and claim time interval,respectively.Firstly,considering the common shock type dependent multi-business risk model,we use the tail risk measure CVaR and policy supervision to constraint the integrated risk,the optimal strategies and the impact of dependence are studied.Through diffusion approximation of the claim risk process,under the condition of normal return of risky assets,the optimal strategy concerning the claim process,investment process,and the proportion of supervisor is obtained,which showes the adverse effect of dependence and the necessity of timely adjusting the premium.Secondly,considering the dependence of claim amount in the homogeneous risk model,we construct optimization modle for an insurer with pairwise quasi-asymptotic independent and regularly-varying-tailed claims,where the price process of risky asset is modled by geometric Brownian motion,the integrated risk was measured by Va R based on maximal discounted net loss in investment period and policy constraint was considered.By using the transforming relationship between the risk measure Va R and the ruin probability,as well as the asymptotic estimate for ruin probability,the approximated optimal strateties are obtained in the case of dangerous claim,the close relationship between the dependence of claim risks and the index of heavy-tail is found.Finally,in view of the dependence between claim amount and claim arrival time interval,the time dependent renewal risk model with FGM dependenc structure is used to characterize the underwriting risk process.Modeling the process of risky asset price with exponential Lévy process,we study the optimal portifolio with VaR constraint.By building the integrated risk model with investment strategy,and basing on uniformly asymptotic estimates for the ruin probability,the approximated optimal strateties are given which concerns the dependence parameters and it is found such a dependence relationship has beneficial effects to insurance company with strong dependence.3.Study on optimal dividend of discrete time risk model with dependent risks.Under the condition that the net loss in different periods of the insurance company is positively dependent through the stochastic ordering,we study the optimal dividend strategies which maximize the expected discounted dividend.Using the dynamic programming principle and state space reduction,the optimal dividend strategies are characterized and the optimality of band strategy is proved.Meanwhile,the properties of value function are discussed,and the numerical algorithm is given.For the konown conclusions involving independence assumption,the corresponding results under dependence condition are given,for some conclusions not involving independence assumption,improvement are made.4.Study on optimal investment and reinsurance with dependent risks to minimize ruin probability.Using common shock type dependent multi-business model to characterize the underwriting risk process,the optimal investment and reinsurance problem are studied,respectively,through diffusion approximation and by applying the dynamic programming principle.Firstly,with market risks including riskless asset and risky asset whose price follows geometric Brownian motion,we get analytic expressions of optimal investment strategy and value function for diffusion model.Combing with numerical examples,we anylze the dynamic effect of dependence intensity,and find that more quantity of risky investment is difficult to compensate the negative effect of risk dependence,the risk of ruin is as well increasing.Secondly,under the variance reinsurance premium principle and arbitary insurance premium principle,we obtain optimal reinsurance strategy concerning with safety loading?claim distribution?counting process and premium rate of insurance,the explicit expression of the minimized ruin probability is also given.Combining expected value insurance premium principle and variance reinsurance premium principle,we find that the optimal retained risk level should increase with rising dependence intensity to prevent more expected loss,as well as the sensitive correlation between the optimal reinsurance proportion and premium rate of insurance.The dependence of risks is currently a topic of general interest and a diffucilty on insurance risk theory and risk management,especially in the case of dependent claim sizes,many classical methods should not be used.This paper takes the dependence of claim risks as the basic point of innovation,and studys the decision problem of optimal reinsurance,optimal investment and optimal dividend in different dependence structures according to seperate background.Furthermore,viewing the objective characterity and reality significance of the criterition of minimizing ruin probability,we especially consider the problem of optimal investment and reinsurance to minimize ruin probability for dependent multi-business model.In addition,when studying the problem of optimal reinsurance,we adopt a uniform optimization criterion,which contains minimizing the variance of retented risks,maximizing the expected concave utility,etc.and the dynamic strategy is also considered;when studying the optimal insurance investment problem,we use tail measure to constraint integrated risk,and the factor of supervisor existing in the investment process of insurance fund is considered;when studying the problem of optimal dividend,we improve parts of the conclusions of dynamic programming principle in discrete time.