Optimal Control Of Dividends And Capital Injections By Reinsurance And Investments

This article studies stochastic control in reinsurance under difusion model. It isdivided into two parts: in the first part, we consider minimizing capital injections underborrowing constraints by reinsurance and investments, discusse the related HJB equa-tions, and finally under three circumstances we give out the equation forms, formalsolution of the optimal strategies, and related numerical solutions. In the second part,we consider maximizing dividends and minimizing capital injections at the same timeby reinsurance and investments, but without borrowing constraints. We focus only ondividends that are paid at a restricted rate, and give out the solution to HJB equations.Since in general cases closed forms are hard to solve, we only give out an integralform of optimal value function, not an explicit closed form. Based on which, we solvethe optimal value function and optimal strategies numerically. At last, under the spe-cial circumstance of cheap reinsurance, we give out the closed form of optimal valuefunction and optimal strategies along with their numerical solutions.