Optimal Investment And Reinsurance Strategies Under Mean-Variance Criteria

Actuarial science is a very natural and important field in the application of stochastic control theory.Among theM,how to determine the optimal investment and reinsurance strategies of insurance companies is not only a stochastic control problem studied by many scholaRs,but also a problem that insurance companies are very concerned about in their actual business.Therefore,the study of optimal investment and reinsurance strategy under different situations based on stochastic control theory not only can enrich and develop stochastic control theory,but also has great application value in the field of actuarial science.In this paper,we mainly study the optimal investment and reinsurance s-trategies of insurance companies,in which the objective criterions are the mean-variance criterions.Specifically,we first study the optimal investment and rein-surance strategies of insurance companies under partial information.In financial practice,investors usually can not directly observe the return information of the risky assets they invest(stocks as an example),but can only get the price in-formation of stocks,which means that decision makers can only make decisions based on partial information.We transform the original problem into a linear quadratic optimal control problem of stochastic systems,in which short selling of stocks is not allowed in the Chinese market,so the control variables are con-strained.Based on the separation principle and stochastic filtering theory,the problem is further transformed into a stochastic linear quadratic optimal con-trol problem with random coefficients.Different from the ordinary differential equations of the Riccati equations,the stochastic Riccati equations we get are the backward stochastic differential equations(BSDEs).Through the solution of BSDEs,we construct the efficient strategies and efficient frontier of the original partial information problem.Next,we consider the optimal investment and reinsurance strategies of insur-ance companies under full information.Similar to the above section,the objec-tive criterion of this optimal control problem is also the mean-variance criterion.Because the mean-variance problem is not a standard stochastic linear quadrat-ic optimal control problem,the general dynamic programming principle is not valid.Based on Lagrange dual theorem,we transform the original problem in-to two sub-problems.First of all,we solve an auxiliary dual problem.Because the control variables are constrained,we construct a solution of the Hamilton-Jacobi-Bellman(HJB)equation in the framework of viscosity solutions.Then,we obtain the efficient strategies and efficient frontier of the full information problem.Finally,we give some numerical examples to verify our theoretical results.Finally,we study a robust optimal investment and reinsurance strategies prob-lem in the case of uncertain model parameters,in which the expected rate of re-turn and volatility of risky assets invested by insurance companies are uncertain parameters.Based on the robust optimization criterion,we transform this prob-lem into a two-person zero-sum stochastic differential game problem on the set of non-equivalent probability measures.Since the objective criterion is the mean-variance criterion,we establish a separation principle through a weak version of the optimality principle and saddle point properties,that is,we can first calculate the minimum value of the risk premium function(the uncertain parameter of the worst case),and then get the optimal strategies of the original problem.Finally,we give some numerical examples to verify our theoretical results.