Research On Ruin Probability And Optimal Control Of Several Kinds Of Risk Models

Risk theory is a very important discipline in insurance mathematics.It is an vital basis for insurance companies to predict and quantitatively analyze the risks caused by various uncertain factors in the course of business operation.How to measure,control,manage,and avoid risks in order to achieve sustained profitability and enhance their competitiveness is an issue that insurers must consider.This paper mainly studies the risk theory of insurance companies,mainly including the following contents:(1)Based on the discrete risk model,considering the influence of the comprehensive interest rate,investment,and reinsurance factors,the bankruptcy problem of the doublebipolar discrete risk model is studied.The properties of the new model are studied by the knowledge of disciplines such as stochastic processes,actuarial calculations,mathematical statistics and numerical simulation.Finally,the Lundberg inequality and its upper bound on the bankruptcy probability of the model are obtained,and the upper bound of the bankruptcy probability of the model is given and the conclusion is analyzed theoretically.Finally,the numerical simulation of the new model is a good test of this conclusion.(2)Aiming at the jump-diffusion risk model,the optimal investment and reinsurance problem that makes insurance companies expect the biggest dividends are studied.Expected premium principle and threshold dividend strategy,we obtain the HJB equation which satisfies the model by using the diffusion approximation theory and the stochastic control theory.Finally,the optimal investment,the optimal reinsurance strategy and the display solution of the expected dividend are obtained.The numerical simulation is used to analyze some parameters impact on the optimal strategy.(3)Based on the jump-diffusion risk model,considering the impact of the surrender and dividend payout,the optimal investment and reinsurance to maximize the expected discounted bonus are studied.The threshold dividend policy is adopted for dividend distribution and the variance premium criterion for premium calculation.The diffusion approximation and stochastic control theory are used to get the HJB equation.Finally,the maximal bonus function and the expression of the optimal investment and reinsurance strategy are obtained,and analyzed the impact of some important parameters on the optimal strategy.(4)Under the jump-diffusion risk model,we study the optimal investment and excess loss reinsurance strategy to maximize the expected benefit of the final value of the surplus.The premium calculation is based on the expectation value premium principle and the stochastic control theory.The HJB equation of the model is obtained finally optimal investment,optimal excess loss reinsurance and explicit solution of value function.The influence of some important parameters on the optimal strategy is analyzed by numerical simulation.