The optimal time to merge for two insurance companies

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 $\mathcal{U}_1$, $\mathcal{U}_2$, $\mathcal{U}_3$, the optimal strategy depends on which subset the reserve processes of the two companies lie in: If they lie in region $\mathcal{U}_1$, the optimal policies are the same as those in Case 1; if they lie in region $\mathcal{U}_3$, the two companies merge immediately and the merged company follows its own optimal dividend policy; if they lie in region $\mathcal{U}_2$, the two companies pay no dividends and wait until their surplus processes reach either region $\mathcal{U}_1$ or region $\mathcal{U}_3$, afterwards, they follow the optimal policies described above.