Minimizing an insurer's probability of ruin with branching businesses

This paper studies an optimization problem for an insurer with several branches. The head insurer can choose a sequence of stopping times to develop new branches. At the same time, the branches have to pay dividends to the head insurer under a predetermined dividend policy. We focus on finding those optimal stopping times to minimize the head company's probability of ruin. By applying the classical optimal stopping theory, the martingale approach, and the induction method, we show the existence of the optimal stopping strategy and identify the value function as the unique viscosity solution of a system of equations. Finally, by applying the comparison theorem, we can provide lower and upper bounds for the minimum probability of ruin and find the estimation error.