Different topological solution structures in a two-dimensional controlled ruin problem depending on the optimization criterion

Abstract

We consider two insurance companies with wealth processes given by independent Brownian motions with controllable non-negative drift. The drift rates sum up to 1. The companies aim at finding a strategy for the drift rates to maximize the probability that at least one of them survives forever. We prove that the strategy, where the company with higher wealth obtains the maximal drift rate, is optimal. Our result differs considerably from the numerical result of McKean and Shepp in [H.P. McKean and L.A. Shepp, The advantage of capitalism vs. socialism depends on the criterion, J. Math. Sci. 139(3) (2006), pp. 6589–6594]. Furthermore, we numerically obtain candidates for the optimal strategy if the common aim of the two companies is to maximize a convex combination of the probability that both firms survive and the probability that exactly one firm survives forever. Our numerical results indicate that in general it is optimal to assign the maximal drift rate to one company but whether it is the company with higher or less wealth depends on the size of both wealth processes.