A continuous-time, consumption/investment problem with constant market coefficients is considered on a finite horizon. A dual problem is defined along the lines of Part 1. The value functions for both problems are proved to be solutions to the corresponding Hamilton-Jacobi-Bellman equations and are provided in terms of solutions to linear, second-order, partial differential equations. As a consequence, a mutual fund theorem is obtained in this market, despite the prohibition of short-selling. If the utility functions are of power form, all these results take particularly simple forms.
"A Duality Method for Optimal Consumption and Investment Under Short-Selling Prohibition. II. Constant Market Coefficients." Ann. Appl. Probab. 2 (2) 314 - 328, May, 1992. https://doi.org/10.1214/aoap/1177005706