We consider the following optimal control problem: Maximize , where X (t) =x+mu t+sigma W (t) -C (t) , tau a parts per thousand inf{t > 0|X (t) =0}a T, T > 0 is a fixed finite time horizon, W (t) is standard Brownian motion, mu, sigma are constants, and C (t) describes accumulated consumption until time t. It is shown that the optimal strategy is given by a barrier strategy with time-dependent continuous barrier b(t). Moreover we determine the asymptotic behavior of b(t) for t -> T.