A ruin problem for a two-dimensional Brownian motion with controllable drift in the positive quadrant

In this paper a two-dimensional Brownian motion (modeling the endowment of two companies), absorbed at the boundary of the positive quadrant, with controlled drift, is considered. We allow that both drifts add up to the maximal value of one. Our target is to choose the strategy in a way s.t. the expected value of the number of surviving companies is maximized. The optimal strategy for this problem is investigated, and it is shown rigorously that the strategy of always pushing maximally the company with less endowment—a strategy which is optimal in the case when one wants to maximize the probability that both companies survive—is in fact not optimal.