Research On Investment Reinsurance In Insurance Market Based On Stochastic Maximum Principle

As an important part of modern control theory,the stochastic optimal control problem has important theoretical value and wide application prospect.Generally speaking,there are two very common and important methods to solve stochastic control problems.One is Bellman's dynamic programming principle.Based on the Bellman optimality,the principle of dynamic programming studies a family of optimal control problems with different initial time and initial states.In the framework of game theory,the relationship between this family of problems and Hamilton-Jacobi-Bellman(HJB)equation,which is a second-order partial differential equation,is established.In the case where HJB equation is solvable,the optimal solution of the problem can be obtained by stochastic verification theorem(SVT).The other is random maximum principle,which can be understood as a random version of Pontryagin maximum principle.Its main content is to give a maximum condition that the optimal control must meet,in order to solve the optimal control problem of stochastic system.This maximum condition is usually described by a Hamilton function defined by a state variable and some dual variables,where the dual variables satisfie one or two backward stochastic differential equations.If appropriate conditions are added,the sufficient conditions of optimal control can also be obtainedThe optimal investment reinsurance problem of insurance companies is a typical stochastic optimal control problem,especially the time-consistent optimal investment reinsurance problem,which has a very important theoretical and practical significance and has attracted the attention of many insurance researchers.When most scholars solve this problem,they usually choose the dynamic programming principle,consider a family of stochastic optimal control problems with different initial time and initial state,establish the relationship with HJB equation in the framework of game theory,and then get the optimal control through SVT.This is mainly because under the mean-variance model,Bellman optimality is invalid,so it can not directly apply to the dynamic programming principle,nor get the time consistent solution.However,there are few articles about the random maximum principle.In fact,on the premise of satisfying the sufficient conditions of the random maximum principle,the strategies obtained by the random maximum principle are also consistent in time.In this thesis,the stochastic maximum principle is used to solve the Stackelberg game between insurance companies and reinsurance companies under the mean-variance criterion.In the second chapter,we consider the stochastic maximum principle under the model related to the wealth process of insurance companies and reinsurance companies,that is to say,the state process is a stochastic process driven jointly by Brownian motion and Poisson martingale measure.The Ito integral and the integral of Poisson martingale measure involved in such a stochastic process can satisfy the equidistant formula,which plays an important role in the deformation of mean-variance.In addition,considering that a more real earnings model of insurance companies is jump diffusion model,which makes insurance companies may have underwriting risk which is related to its investment risk.Therefore,in the third section,we consider the stochastic differential equation driven jointly by a two-dimensional Brownian motion with relevance and compound Poisson stochastic process,and solve the problem of relevance by the levy characterization of martingales.However,it should be noted that the stochastic maximum principle only gives the possibility of the optimal strategy.Therefore,in the second section,the sufficient conditions of the stochastic maximum principle are added to ensure that the optimal solution we get in the third chapter is sufficient and necessary.In the third chapter,we apply the theory of stochastic maximum principle to proportional reinsurance and excess loss reinsurance respectively,discuss the time consistent strategy of insurer and reinsurer,and discuss the Stackelberg Equilibrium Strategy of both sides under appropriate restrictions.