Mathematics – Optimization and Control
Scientific paper
2012-03-16
Mathematics
Optimization and Control
25 pages
Scientific paper
In this paper we study a continuous time, optimal stochastic investment problem under limited resources in a market with N firms. The investment processes are subject to a time-dependent stochastic constraint. Rather than using a dynamic programming approach, we exploit the concavity of the profit functional to derive some necessary and sufficient first order conditions for the corresponding Social Planner optimal policy. Our conditions are a stochastic infinite-dimensional generalization of the Kuhn-Tucker Theorem. As a subproduct we obtain an enlightening interpretation of the first order conditions for a single firm in Bank [5]. In the infinite-horizon case, with operating profit functions of Cobb-Douglas type, our method allows the explicit calculation of the optimal policy in terms of the base capacity process, i.e. the unique solution of the Bank and El Karoui representation problem [4].
Chiarolla Maria B.
Ferrari Giorgio
Riedel Frank
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