We construct I mu-optimal strategies for the following 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 I mu-optimal strategies are given by barrier strategies with time-dependent barriers.