A duality method for optimal consumption and investment under short-selling prohibition

Abstract: "A continuous-time, consumption/investment problem on a finite horizon is considered for an agent seeking to maximize expected utility from consumption plus expected utility from terminal wealth. The agent is prohibited from selling stocks short, so the usual martingale methods for solving this problem do not directly apply. A dual problem is posed and solved, and the solution to the dual problem provides information about the existence and nature of the solution to the original problem. When the market coefficients are constant, the value functions for both problems are provided in terms of solutions to linear, second-order, partial differential equations. If, furthermore, the utility functions are of the power form, the solutions to these equations take a particularly simple form, as do the formulas for the optimal consumption and investment processes."