两个保险公司的最优合并时刻问题

The optimal time to merge for two insurance companies

Gerber和Shiu(2006)将两个公司的合并可行性问题与最优分红策略问题联系起来,给出公司合并会产生效益的条件,并提出一个更现实的问题:若选择合并,最优合并时刻是什么?本文分两步研究了这个问题.首先,利用混合奇异/二维最优停止理论,证明相应的验证定理,然后分两种情形讨论了这个问题.最终得到的最优策略为:情形1下不可能合并,两个公司分别采取最有利于自己的分红策略;情形2下,整个区域被分成3部分U_1、U_2和U_3,最优策略取决于两公司的盈余落在哪个区域.若落在区域U_1,最优策略同情形1;若落在区域U_3,两公司马上合并,合并后的公司采取它的最优分红策略;若落在区域U_2,两公司不分红并等待,直到其盈余过程到达U_1或U_3,然后,施行如上所述的合并和分红策略.

In Gerber and Shiu(2006)an interesting link is provided between the feasibility of a merger of two companies and the theory of optimal dividend strategy.They give a situation in which the merger of the two companies will result in a gain and this result can give a useful guideline on corporate governance.At the end of Gerber and Shiu(2006),a more realistic problem is raised,i.e.,what is the best time to merge?In order to analyze this problem,we first use the theory of the mixed singular control/two-dimensional optimal stopping to prove the corresponding dynamic programming principle and the verification theorem.Then,we consider the problem in two cases.In Case 1,merging of companies is never necessary and the two companies pay dividends according to their own optimal dividend policies.In Case 2,the problem is much more difficult.We split the whole region into three subsets U1,U2,U3,the optimal strategy depends on which subset the reserve processes of the two companies lie in:If they lie in region U1,the optimal policies are the same as those in Case 1;if they lie in region U3,the two companies merge immediately and the merged company follows its own optimal dividend policy;if they lie in region U2,the two companies pay no dividends and wait until their surplus processes reach either region U1or region U3,afterwards,they follow the optimal policies described above.