Optimal Management Decisions Of Insurance Companies Under Game Theory

With the development of the insurance market,risk management has become important in the actuary study.Insurance companies are mainly faced with two aspects of risk,one is the claim risk from the insurance market,and the other is the investment risk from the financial market.This paper studies the optimal premium,investment,and reinsurance strategies of insurance companies with the game theory,including non zero sum game problems between two insurance companies,and Stackelberg game between reinsurance company and insurance companies.Based on risk theory,utility theory,and stochastic control theory,the problems are solved to obtain the optimal equilibrium strategies,providing important theoretical support for managers to make optimal decisions.This study mainly includes the following two parts:Firstly,we study the optimal investment,reinsurance and premium problem in a nonzero-sum game between two competing insurers and find the Nash equilibrium strategies under the time-consistent mean-variance criterion.We formulate the insurer’s claim process into a non-homogeneous compound Poisson process and assume that the safety loading and the claim arrival rate are connected through a monotone decreasing function.The insurers invest the surplus in one risk-free asset and one risky asset described by the Heston model.By solving the extended HJB equations,we obtain the equilibrium strategies and the corresponding equilibrium value functions.Furthermore,some numerical analyses are conducted to show the effect of model parameters on the equilibrium strategies.Secondly,we study the reinsurance-investment problem among one reinsurer and two insurers with ambiguity aversion in a hybrid game,where the reinsurer is the leader and two insurers are the followers in a Stackelberg game,and the two competitive insurers are also involved in a non-zero-sum game.We also investigate how ambiguity aversion and competitive relationship influence the optimal reinsurance-investment strategies.we consider the mean-variance criterion,that is,maximizing the expectation and minimizing the variance of terminal wealth as well.Finally,we show some numerical experiments to demonstrate the effects of some parameters on the strategies.