Research On Robust Investment-reinsurance Strategies Of The Ambiguity-averse Insurer

In this paper,we consider several classical risk models,the price's process of risk assets and the surplus process of insurers by some mathematical tools,such as stochastic control theory,stochastic analysis,dynamic programming and so forth.We assume that an ambiguity-averse insurer(AAI)has a reference model,but due to the randomness and doubt of the AAI,she hopes to consider a group of alternative models characterized by a family of equivalent measures.The penalty function is used to describe the deviation between the reference model and the alternative models.When the deviation is larger,the greater the penalty the reference model receives,the greater the uncertainty of the model,the insurer need a robust investment-reinsurance strategy in this worst case.From the standpoint of insurers,so we study some robust investmentreinsurance problems of these models under uncertainty in Chapters 3-7.The main researches are as follows.At first,in Chapter 3,we consider the optimal reinsurance and investment problem under the jump-diffusion risk model,in which the insurer calculates the premium income based on the expected premium principle.But,the premium calculation of the reinsurer is based on the generalized mean-variance premium principle,and the form of reinsurance is proportional reinsurance.The stock's price is described by a constant elasticity of variance(CEV)model.For an AAI,the parameters of the model are uncertainty.We use the relative entropy penalty to describe the deviation between the model and the real model,and use dynamic programming principle to deduce the explicit expressions of robust optimal investment-reinsurance strategies aiming at maximizing the expected terminal utility,and the value function,and give the verification theorem.Some numerical examples show the influences of the parameters on investment-reinsurance strategy,these results combined with economic phenomenon are analyzed.Secondly,in Chapter 4,we study the robust optimal reinsurance-investment under the jump-diffusion risk model,in which the premium of reinsurer is based on the generalized mean-variance premium principle.Different from Chapter3,the form of reinsurance is based on a self-reinsurance function,including proportional reinsurance and excess-loss reinsurance.The stock's price process is described by jump-diffusion model.For an AAI,the model parameters are uncertainty,and the jump size and jump intensity of the stock's price are also uncertainty.The deviation between the model and the real model is described by the relative entropy penalty.Using the principle of dynamic programming,we deduce the expressions of robust optimal investment-reinsurance strategies,prove the verification theorem.Some numerical examples are used to show the influence of parameters on investment strategy,these economic phenomenons are analyzed.Thirdly,in Chapter 5,we consider the non zero-sum game problem of two ambiguity-averse insurers.We assume that the surplus processes of the two insurers are two jump-diffusion models with common shock.They invest in the same bank,the same stock,the same credit default swap(CDS),and buy the insurance from different reinsurance companies to transfer some of their risk.Based on the principle of dynamic programming,the robust Nash equilibrium investment strategies of the two insurers are obtained and proved in detail.Some numerical results verify our theoretical results.Fourthly,in Chapter 6,we assume that the AAI can invest in risk-free assets,stocks,and defaultable corporate bond and purchase reinsurance.The price process of stock is described by jump-diffusion process,and the surplus process is described by the classic C-L process.Assuming that the stock's jump and the claim's jump have a common impact.The model is uncertainty.Different from Chapter 3-5,we consider the form of the penalty as the generalized penalty function.The game between the insurer and the market is discussed.Based on the principle of dynamic programming,when the form of penalty function is quadratic and linear,we get the explicit solutions of the robust optimal investment-reinsurance strategies of insurer,and analyze the influence of different parameters through some numerical examples.Finally,in Chapter 7,we study the non zero-sum game of two ambiguityaverse insurers according to Chapter 5 and 6,and assume that the surplus processes of the two insurers are the classic C-L models.The AAI can invest in the same bank,the same stock,the same kind of defaultable corporate bond,and buy reinsurance from different reinsurance companies to transfer some of their risks.We consider the form of the penalty as the generalized penalty function,two insurers compete with each other and play games with the market.Based on the principle of dynamic programming,the form of penalty function is quadratic and linear,we get the robust Nash equilibrium investment strategies of the two insurers.Some numerical results verify our theoretical results.