Before revision:  We learn that, for wealth lying between 0 and b , the optimal investment strategy is {\it independent } of b , a surprising result.
After revision:  We learn that, for wealth lying between 0 and b , a type of goal-seeking problem, as pioneered by Dubins and Savage (1965, 1976). The individual consumes at a constant rate c, so the level of wealth required for risklessly meeting consumption equals c/r, in which r is the rate of return of the riskless asset. Our problem is related to, but different from, the goal-reaching problems of Browne (1997). First, Browne (1997, Section 3.1) maximizes the probability that wealth reaches b < c/r before it reaches a < b. Browne's game ends when wealth reaches b. By contrast, for the problem we consider, the game continues until the individual dies or until wealth reaches 0; reaching b and then falling below it before death does not count. Second, Browne (1997, Section 4.2) maximizes the expected discounted reward of reaching b > c/r before wealth reaches c/r. If one interprets his discount rate as a hazard rate, then our two problems are {\it independent } of b , a surprising result.