Nonlinear Optimal Control Problem Of An Insurance Company Under Solvency Constraints

This paper considers a dividend optimization problem for an insurance companywhich can accommodate the profit and the risk by adjusting its reinsurance rate andoperating its dividend payout process at any time before bankruptcy. The main goalwe aim at is to find the optimal policy so as to maximize the expected present valueof the dividend until the time of bankruptcy. We will pick up the classical profit andloss model which can be improved as follows. Firstly, the optimal policy may be soextreme that will lead the company result in bankruptcy soon. So we do need to addsome solvency constraints to this problem. In addition, taking safety and profits intoconsideration, the management of reinsurer has to impose its preferred risk level andadditional charge on firm seeking services beyond or lower than the level. This turnsout to be a nonlinear optimal stochastic control problem with positive reinsurance costunder solvency constraints.We work out this problem by building up the HJB equation so as to solve theoptimal control policy of the insurance company. Particularly we show that there existsan optimal barrier b such that the optimal dividend policy is to keep the company'swealth below b and pay out as dividends all the amounts in excess of this barrier.we will give a complete proof to show the existence and uniqueness of the minimumdividend barrier. The optimal return function associated with the optimal policy willbe given at last. We will give an economic explanation for our main result,and thengive a quantitative analysis about the optimal dividend problem and economic capitalof the insurance company.Solving this problem required the knowledge of financial mathematics, insur-ance,stochastic analysis and optimization.The conclusion in this paper will play a rolein guiding insurance industry development.