Optimal Dividend And Capital Injection Problem Of The Nonlinear Risk Process With Proportional Cost

In recent years, the optimal dividend, finance and reinsurance has become a hot issue in the actuarial research, which has a detailed analysis at the micro level on the reinsurance arrangements and investments appearing in the process of the risk exchanging and transferring so that the insurance company in operation could obtain the higher profits, greater utility and lower risks, etc.This paper considers the optimal dividend payments and capital injection control problem in a nonlinear risk process attributed to internal competition factors. We also incorporate other realistic elements such as the boundary of the regular control variables, rate of the debt repayment, proportional cost, which lead to the process being closer to the realistic situation. Our objective is to maximize the expectation of the cumulative discounted dividend payments minus the discounted costs of capital injection before bankruptcy. This problem is formulated as a nonlinear mixed singular-regular stochastic control problem. Considering whether there is a range limitation on the reinsurance policies, the optimal problem can be divided into two kinds of situations. One is that there is no upper bound on the reinsurance policies. The other is that the changes of the reinsurance policies are limited in some a fixed range.In the first case, where the reinsurance policies in the nonlinear risk model satisfies Uπ(t)>0, by discussing that how to obtain the optimal control policy and then solving the corresponding HJB equations, the specific form of value function is obtained and the optimal strategy is a mixed singular-regular stochastic control strategy, respectively.In the second case, where the reinsurance policies in the nonlinear risk model satisfies Uπ(t)∈[l, u], by analyzing the optimal control strategies in the six different cases as the result of the intersection of two sets ([l, u] and [M+2δ/μ,μ/2a]) and then solving the corresponding HJB equations, the specific form of value function and the corresponding optimal control strategies are given. At last, when there is no debt rate in the model, the exact analytic type of value function in this special case is obtained.