| In this paper we consider optimal control problem of a large insurance companyin which the dividend pay-outs, investing process and the risk exposure are controlledby management. The company has a minimal reserve as its guarantee fund to protectinsureds and attract su?cient number of policy holders. The company is consideredbankrupt as soon as the reserve falls below m. We assume that the company can onlyreduce its risk exposure by proportional reinsurance policy for simplicity. The objectiveis to find a optimal policyπ* = {aπ*(t), Ltπ*}, consisting of risk control and dividendpayment scheme, which maximizes the expected total discounted dividend pay-outsuntil the time of bankruptcy. This paper aims at resolving the optimal control problemof a large insurance company under higher standard of solvency and security and tofind the best equilibrium policy between making profit and improving security.We confines standard of solvency and security by controlling the dividend bound-ary b,and obtain the sub-optimal return function V(x, b) as well as the actual sub-optimal policyπb*.We let standard of solvency be that the probability of the insur-ance company with initial reserve b going bankrupt in time horizon T can not excessε> 0,that is P[τbπb*≤T]≤ε. objective is to find a suitable dividend boundary b?which is in B := b : P[τbb≤T]≤ε,such that V(x, b*) = supb∈B{V(x, b)}.We willprove the properties of probability of bankruptcy such as continuity ,monotone de-creasing and Converge towards zero as b→0.We get the monotone decreasing prop-erty of sub-optimal return function V(x, b) with b ,implying that b* = min B such thatV(x, b*) = supb∈B{V(x, b)}.The innovation of this paper is the discussion of the bankruptcy boundarym,proving that the probability of bankruptcy has a lower boundary greater than 0,im-plying the significance of the research of solvency. |
