Optimally investing to reach a bequest goal

We determine the optimal strategy for investing in a Black–Scholes market in order to maximize the probability that wealth at death meets a bequest goal bb, a type of goal-seeking problem, as pioneered by Dubins and Savage (1965, 1976). The individual consumes at a constant rate cc, so the level of wealth required for risklessly meeting consumption equals c/rc/r, in which rr 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 bc/rb>c/r before wealth reaches c/rc/r. If one interprets his discount rate as a hazard rate, then our two problems are mathematically   equivalent for the special case for which b>c/rb>c/r, with ruin level c/rc/r. However, we obtain different results because we set the ruin level at 0, thereby allowing the game to continue when wealth falls below c/rc/r.