  In this paper, we consider the optimal dividends problem for a company whose
cash reserves follow a general Levy process with certain positive jumps and
arbitrary negative jumps. The objective is to find a policy which maximizes the
expected discounted dividends until the time of ruin. Under appropriate
conditions, we appeal to very recent results in the theory of potential
analysis of subordinators to obtain the convexity properties of probability of
ruin. We present conditions under which the optimal dividend strategy, among
all admissible ones, takes the form of a barrier strategy.
