Optimal Investment And Reinsurance Strategies With Credit Risk Under The Framework Of Game Theory

Insurance companies manage their wealth by investing in financial markets,and the problem of optimal investment is one of the main issues studied by actuarial practitioners and researchers.In addition,how to spread risk through reinsurance is particularly important for insurance companies.At present,many papers focus on the optimal investment and reinsurance of single insurance company,but the business competition and cooperation among various companies in the insurance market are complicated.Naturally,the game problem among insurance companies deserves wide attention.In addition,as a typical fixed income investment,bonds are the most favored investment products of insurance companies and reinsurance companies.In this paper,we study the game of insurance market in different environments while considering credit risk.It mainly includes non-zero-sum game and Stackelberg game between insurance companies and reinsurance companies,non-zero-sum game between insurance companies and two-layer game between a reinsurance company and two insurance companies.We use stochastic control theory,risk theory,game theory and so on to model these problems and calculate the equilibrium strategy,so as to provide important basis for the company to make decisions.This paper mainly includes the following four parts:Firstly,this paper studies the non-zero-sum investment and reinsurance game between insurance companies and reinsurance companies on the premise of considering credit risk.Our starting point is the cooperative game between insurance companies and reinsurance companies.Assuming that both companies can invest their surplus in risk-free assets,risky assets and defaultable bonds,a cooperative game framework is constructed for the two companies.Based on the stochastic dynamic programming method,we obtain the HJB equation of the optimization problem and give the equilibrium investment strategy,and analyze the influence of cooperation intensity on the equilibrium strategy.In addition,we analyze the pricing problem in the reinsurance market according to the principle of supply and demand.Compared with the traditional pricing method,this method is more in line with the development law of the reinsurance market.Secondly,based on the leader-follower relationship between reinsurance companies and reinsurance companies,the Stackelberg investment and reinsurance game model of the two companies is constructed under the premise of considering credit risks.As the leader of the Stackelberg game,the reinsurance company first determines the reinsurance price and investment strategy.As the follower of the game,the insurance company can determine the proportion of reinsurance and its own investment strategy according to the price of reinsurance premium.Both insurers and reinsurers can invest their wealth in risk-free assets,defaultable bonds and risky assets whose price processes satisfy the Heston model.In addition,we consider the effect of the delay of the wealth process on the equilibrium strategy.Stackelberg equilibrium strategy and value function are derived explicitly by backward induction method and dynamic programming method,and parameter sensitivity analysis of the equilibrium strategy is carried out by numerical simulation.Thirdly,on the premise of considering credit risk,this paper uses relative performance as the measurement standard of competition mechanism under the background of model uncertainty and limited memory characteristics,and studies how insurance companies choose the optimal strategy in the increasingly fierce competition environment.We assume that the two insurance companies with ambiguity aversion can invest in risk-free assets,defaultable bonds and risk assets whose price process satisfies the Heston model.We construct a non-zero-sum game model for the two insurance companies,and establish the HJBI equation of this problem by using the principle of dynamic programming,and obtain the robust optimal investment and reinsurance strategy.Finally,the influence of model parameters on robust optimal strategy is demonstrated by numerical analysis.Finally,this paper extends the game model mentioned above and constructs a two-layer stochastic differential game model involving three companies.The first layer of the two-layer game model is Stackelberg game between the reinsurance company and two insurance companies,and the second layer is non-zero-sum game between two competing insurance companies.Suppose an insurance company can invest in risk-free assets,defaultable bonds,and risky assets characterized by bi-fraction Brownian motion,and a reinsurance company invests in risk-free assets and risky assets characterized by another bi-fraction Brownian motion environment.By solving the corresponding HJB equation,Nash equilibrium solution and Stackelberg equilibrium strategy are obtained,and the influence of model parameters on equilibrium strategy is explained by numerical analysis.