Optimal consumption and arbitrage in incomplete, finite state security markets

The authors study a consistent treatment for both the multi-period portfolio selection problem and the option attainability problem by a dual approach. They assume that time is discrete, the horizon is finite, the sample space is finite and the number of securities is less than that of the possible securities price transitions, i.e. an incomplete security market. The investor is prohibited from investing stocks more than given linear investment amount constraints at any time and maximizes an expected additive utility function for the consumption process. First the authors give a set of budget feasibility conditions so that a consumption process is attainable by an admissible portfolio process. To establish this relation, they used an algorithmic approach which has a close connection with the linear programming duality. Then the authors prove the unique existence of a primal optimal solution from the budget feasibility conditions. Finally, they formulate a dual control problem and establish the duality between primal and dual control problems.