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.
"A Duality Method for Optimal Consumption and Investment Under Short- Selling Prohibition. I. General Market Coefficients." Ann. Appl. Probab. 2 (1) 87 - 112, February, 1992. https://doi.org/10.1214/aoap/1177005772