Actions on Environment under uncertainty: stochastic formulation and the associated deterministic problem

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Abstract

An application of the results of this paper proves that there is not always an economic benefit when destroying the environment for planting an alternative industrial project. Our criterion, to act, to delay or to deny the industrial investment over the environment, is given in terms of the free boundary associated to a deterministic degenerate obstacle problem (on an unbounded domain) associated to the stochastic optimal control problem formulated, initially, in terms of some suitable stochastic diffusion processes. The localizing estimates on the free boundary are obtained through a suitable spatial change of variables and by working with a suitable distance associated to the coefficient of the elliptic operator.

Keywords

degenerate obstacle problem unbounded domain stochastic optimal control problem environmental economy 

Mathematics Subject Classifications

35T35 60H15 93E20 91B76 

Acciones sobre el Medio Ambiente bajo incertidumbre: formulaicón Estocástica y el problema determinista asociado

Resumen

Una interpretación de los resultados de este trabajo muestra que no siempre hay beneficio Económico cuando se destruye el medio ambiente para la implantación de un proyecto industrial alternativo. Nuestro criterio, de actuar, retrasar o negar la inverseón industrial sobre el medio ambiente, viene dado en términos de la frontera libre asociada a un problema de obstáculo determinista degenerado (sobre un dominio no acotado) asociado al problema estocástico de control óptimo formulado, inicialmente, en términos de ciertos procesos de difusión estocásticos.

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References

  1. [1]
    Akerlof, G. A. (1998), The Market for Lemons: Qualitative Uncertainty and the Market Mechanism,Quarterly Journal of Economics,84, 488–500.CrossRefGoogle Scholar
  2. [2]
    Bellman, R. (1957),Dynamic Programming. Academic Press, London.MATHGoogle Scholar
  3. [3]
    Bensoussan, A. andLions, J. L., (1978).Application des inégalités variationnelles en control stochastique. Dunod, Paris.Google Scholar
  4. [4]
    Bermudez, A., Moreno, C. and Sanmartin, A., (1997). Resolución numérica de un problema de valor óptimo de una opción. InActas de la Jornada Científica en homenaje a A. Valle, Caraballo, T., et al Eds, Publicaciones de la Universidad de Sevilla.Google Scholar
  5. [5]
    Brezis, H., (1972). Problémes Unilateraux.J. Math. Pures et Appl.,51, 1–168.MathSciNetGoogle Scholar
  6. [6]
    Brezis, H., (1983).Analyse fonctionnelle. Masson, Paris.MATHGoogle Scholar
  7. [7]
    Brezis, H. andFriedman, A., (1976). Estimates on the support of the solutions of parabolic variational inequalities.Illinois J. Math,20, 82–97.MATHMathSciNetGoogle Scholar
  8. [8]
    Crandall, M. G., Ishii, H. andLions, P. L., (1992). User’s guide to viscosity solutions of second order partial differential equations.Bull. Amer. Math. Soc.,27, 1–67.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Díaz, G., (1985). Acción óptima en una ecuación de la Programación Dinámica.Rev. R. Acad. Cienc. Exactas Fís. Nat.,LXXIX, 89–105.Google Scholar
  10. [10]
    Díaz, G., Díaz, J. I., Faghloumi C., (2007). On an evolution problem associated to the modelling of incertitude into the Environment.Nonlinear Analysis: Real World Applications,8, 399–404.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    Díaz, J. I., (1985).Nonlinear partial differential equations and free boundaries.106, Pitman, London.MATHGoogle Scholar
  12. [12]
    Díaz, J. I., Faghloumi, C., (2002). Analysis of a nonlinear elliptic problem arising in the study of policies on projects altering the environment,Applied Math. and Optimization,45, 251–267.MATHCrossRefGoogle Scholar
  13. [13]
    Dixit, A. K. andPindyck, R. S., (1994).Investment under Uncertainly. Princeton University Press, Princeton.Google Scholar
  14. [14]
    Faghloumi, C., (2004).Modelizacion y tratamiento de algunos problemas de Medio Ambiente. Ph. D.,thesis, Universidad Complutense de Madrid, Madrid.Google Scholar
  15. [15]
    Fleming, W. H. andRishel, R. (1975).Deterministic and stochastic optimal control, Springer-Verlag, New York.MATHGoogle Scholar
  16. [16]
    Herbelot, O., (1992).Option Valuation of Flexible Investments: The Case of Environmental Investments in the Electric Power Industry, Ph. D., M.I.T., Massachusetts.Google Scholar
  17. [17]
    Kinderlherer, D. andStampacchia, G., (2000).An introduction to Variational Inequalities and its Applications, Second edition, SIAM, Philadelphia.Google Scholar
  18. [18]
    Lions, J. L., (1969).Quelques méthodes de resolution des problémes aux limites non lineaires, Dunod, París.MATHGoogle Scholar
  19. [19]
    Oksendal, B., (1998).Stochastic Differential Equations. Springer 5th edition, Berlin.Google Scholar
  20. [20]
    Pindyck, R. S., (1986). Irreversible Investments, Capacity Choice and the Value of the Firm.American Economic Review,78, 707–728.Google Scholar
  21. [21]
    Scheinkman, J. A., (1994).Public goods and the Environment, in Environment Economics and their Mathematical Models, Díaz, J. I. and Lions, J. L. Eds, Masson, Paris, 149–158.Google Scholar
  22. [22]
    Troianiello, G. M., (1987).Elliptic differential equations and obstacle problems. Plenum Press, New York.MATHGoogle Scholar

Copyright information

© Springer 2008

Authors and Affiliations

  1. 1.Dpto. de Matemática Aplicada Facultad de MatemáticasUniversidad Complutense de MadridMadridSpain
  2. 2.Bayes Inference S. A.MadridSpain

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