2015/03/03 by Bayraktar, Erhan, Young, Virginia R.
#FOS: Economics and business #FOS: Mathematics #Mathematical Finance (q-fin.MF) #Optimization and Control (math.OC) #Portfolio Management (q-fin.PM) #Probability (math.PR)
paper · doi:10.48550/arxiv.1503.00961
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 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 mathematically equivalent for the special case for which b > c/r, with ruin level c/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/r.