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Viscosity solutions approach to economic models governed by DDEs

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  • Fabbri, Giorgio

Abstract

A family of economic and demographic models governed by linear delay differential equations is considered. They can be expressed as optimal control problems subject to delay differential equations (DDEs) characterized by some non-trivial mathematical difficulties: state/control constraints and delay in the control. The study is carried out rewriting the problem as an (equivalent) optimal control problem in infinite dimensions and then using the dynamic programming approach (DPA). Similar problems have been studied in the literature using classical and strong (approximating) solutions of the Hamilton-Jacobi-Bellman (HJB) equation. Here a more general formulation is treated thanks to the use of viscosity solutions approach. Indeed a general current objective function is considered and the concavity of the Hamiltonian is not required. It is shown that the value function is a viscosity solution of the HJB equation and a verification theorem in the framework of viscosity solutions is proved.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 2826.

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Date of creation: 2006
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Handle: RePEc:pra:mprapa:2826

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Keywords: viscosity solutions; delay differential equation; vintage models;

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  1. Raouf Boucekkine & David De La Croix & Omar Licandro, 2004. "MODELLING VINTAGE STRUCTURES WITH DDEs: PRINCIPLES AND APPLICATIONS," Mathematical Population Studies, Taylor and Francis Journals, vol. 11(3-4), pages 151-179.
  2. Silvia Faggian & Fausto Gozzi, 2004. "On The Dynamic Programming Approach For Optimal Control Problems Of Pde'S With Age Structure," Mathematical Population Studies, Taylor and Francis Journals, vol. 11(3-4), pages 233-270.
  3. Asea, Patrick K. & Zak, Paul J., 1999. "Time-to-build and cycles," Journal of Economic Dynamics and Control, Elsevier, vol. 23(8), pages 1155-1175, August.
  4. Boucekkine, Raouf & Del Rio, Fernando & Licandro, Omar, 2000. "Vintage capital and the dynamics of the AK model," CEPREMAP Working Papers (Couverture Orange) 0003, CEPREMAP.
  5. Boucekkine, Raouf & Licandro, Omar & Paul, Christopher, 1997. "Differential-difference equations in economics: On the numerical solution of vintage capital growth models," Journal of Economic Dynamics and Control, Elsevier, vol. 21(2-3), pages 347-362.
  6. Mauro Bambi, 2006. "Endogenous growth and time to build: the AK case," Computing in Economics and Finance 2006 77, Society for Computational Economics.
  7. Raouf Boucekkine & Marc Germain & Omar Licandro & Alphonse Magnus, . "Numerical solution by iterative methods of a class on vintage capital models," Working Papers 97-26, FEDEA.
  8. Boucekkine, Raouf & de la Croix, David & Licandro, Omar, 2002. "Vintage Human Capital, Demographic Trends, and Endogenous Growth," Journal of Economic Theory, Elsevier, vol. 104(2), pages 340-375, June.
  9. Gustav Feichtinger & Richard F. Hartl & Suresh P. Sethi, 1994. "Dynamic Optimal Control Models in Advertising: Recent Developments," Management Science, INFORMS, vol. 40(2), pages 195-226, February.
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