Optimal investment models with vintage capital: dynamic programming approach. (English) Zbl 1196.49022
Summary: The dynamic programming approach for a family of optimal investment models with vintage capital is developed. The problem falls into the class of infinite horizon optimal control problems of PDEs with age structure that have been studied in various papers [E. Barucci and F. Gozzi, Investment in a vintage capital model, Res. Econ. 52, 159–188 (1998); J. Econ. 74, No. 1, 1–38 (2001; Zbl 1026.91073); G. Feichtinger, G. Tragler and V. M. Veliov, J. Math. Anal. Appl. 288, No. 1, 47–68 (2003; Zbl 1042.49035) and G. Feichtinger, R. F. Hartl, P. M. Kort and V. M. Veliov, J. Econ. Theory 126, No. 1, 143–164 (2006; Zbl 1108.91055)] either in cases when explicit solutions can be found or using maximum principle techniques.
The problem is rephrased into an infinite dimensional setting, it is proven that the value function is the unique regular solution of the associated stationary Hamilton-Jacobi-Bellman equation, and existence and uniqueness statements of optimal feedback controls are derived. It is then shown that the optimal path is the solution to the closed loop equation. Similar results were proven in the case of finite horizon by S. Faggian [Appl. Math. Optimization 51, No. 2, 123–162 (2005; Zbl 1078.49026) and Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 15, No. 4, 527–553 (2008; Zbl 1223.49030)]. The case of infinite horizon is more challenging as a mathematical problem, and indeed more interesting from the point of view of optimal investment models with vintage capital, where what mainly matters is the behavior of optimal trajectories and controls in the long run.
Finally it is explained how the results can be applied to improve the analysis of the optimal paths previously performed by Barucci and Gozzi and by Feichtinger et al.
The problem is rephrased into an infinite dimensional setting, it is proven that the value function is the unique regular solution of the associated stationary Hamilton-Jacobi-Bellman equation, and existence and uniqueness statements of optimal feedback controls are derived. It is then shown that the optimal path is the solution to the closed loop equation. Similar results were proven in the case of finite horizon by S. Faggian [Appl. Math. Optimization 51, No. 2, 123–162 (2005; Zbl 1078.49026) and Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 15, No. 4, 527–553 (2008; Zbl 1223.49030)]. The case of infinite horizon is more challenging as a mathematical problem, and indeed more interesting from the point of view of optimal investment models with vintage capital, where what mainly matters is the behavior of optimal trajectories and controls in the long run.
Finally it is explained how the results can be applied to improve the analysis of the optimal paths previously performed by Barucci and Gozzi and by Feichtinger et al.
MSC:
| 49L20 | Dynamic programming in optimal control and differential games |
| 49J27 | Existence theories for problems in abstract spaces |
| 91G10 | Portfolio theory |
Keywords:
optimal investment; vintage capital; age-structured systems; optimal control; dynamic programming; Hamilton-Jacobi-Bellman equations; linear convex control; boundary controlReferences:
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