Optimal Investment And Proportional Reinsurance Problem With Common Shock Dependence

Optimal investment and proportional reinsurance problem has long been an im-portant topic in the actuarial science.In this thesis,from the perspective of the in-surer,we assume the insurer can invest its wealth in the financial market at any time and the portfolio includes a risky asset and a riskless asset.Under the risk model with two dependent classes of insurance business,where the two claim number process-es are correlated through a common shock component,we study two optimization problems.One problem is maximizing the expectation of the discounted time to reach a goal,namely,the insurer expects its wealth to reach a given goal as early as possi-ble;the other problem is minimizing the probability of drawdown,which means the value of the surplus process reaches some fixed proportion of its maximum value to date.Since the two dependent classes of insurance business of the insurer are mod-ulated by compound Poisson model,we use diffusion approximation risk model to describe the insurer's surplus process.Based on the technique of stochastic control theory and the corresponding Hamilton-Jacobi-Bellman equation,firstly,we provide a verification lemma,which shows the classical solution of a boundary-value prob-lem(BVP)equals the maximium expectation of the discounted time to reach a goal.Therefore,we can replace our problem by the one of solving a BVP.Restraining the retention levels of the insurer's claims in the interval[0,?),we discuss the solution of the BVP and the optimal strategies in different situations.Furthermore,we con-strain the retention levels of the insurer's claims in the interval[0,1],by the method of Hamilton-Jacobi-Bellman equation,we derive the explicit expressions of the opti-mal investment and reinsurance strategies and the minimum probability of drawdown.Finally,some numerical examples are presented to show the impact of model param-eters on the optimal results.