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.
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.
Similar content being viewed by others
References
Akerlof, G. A. (1998), The Market for Lemons: Qualitative Uncertainty and the Market Mechanism,Quarterly Journal of Economics,84, 488–500.
Bellman, R. (1957),Dynamic Programming. Academic Press, London.
Bensoussan, A. andLions, J. L., (1978).Application des inégalités variationnelles en control stochastique. Dunod, Paris.
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.
Brezis, H., (1972). Problémes Unilateraux.J. Math. Pures et Appl.,51, 1–168.
Brezis, H., (1983).Analyse fonctionnelle. Masson, Paris.
Brezis, H. andFriedman, A., (1976). Estimates on the support of the solutions of parabolic variational inequalities.Illinois J. Math,20, 82–97.
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.
Díaz, G., (1985). Acción óptima en una ecuación de la Programación Dinámica.Rev. R. Acad. Cienc. Exactas Fís. Nat.,LXXIX, 89–105.
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.
Díaz, J. I., (1985).Nonlinear partial differential equations and free boundaries.106, Pitman, London.
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.
Dixit, A. K. andPindyck, R. S., (1994).Investment under Uncertainly. Princeton University Press, Princeton.
Faghloumi, C., (2004).Modelizacion y tratamiento de algunos problemas de Medio Ambiente. Ph. D.,thesis, Universidad Complutense de Madrid, Madrid.
Fleming, W. H. andRishel, R. (1975).Deterministic and stochastic optimal control, Springer-Verlag, New York.
Herbelot, O., (1992).Option Valuation of Flexible Investments: The Case of Environmental Investments in the Electric Power Industry, Ph. D., M.I.T., Massachusetts.
Kinderlherer, D. andStampacchia, G., (2000).An introduction to Variational Inequalities and its Applications, Second edition, SIAM, Philadelphia.
Lions, J. L., (1969).Quelques méthodes de resolution des problémes aux limites non lineaires, Dunod, París.
Oksendal, B., (1998).Stochastic Differential Equations. Springer 5th edition, Berlin.
Pindyck, R. S., (1986). Irreversible Investments, Capacity Choice and the Value of the Firm.American Economic Review,78, 707–728.
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.
Troianiello, G. M., (1987).Elliptic differential equations and obstacle problems. Plenum Press, New York.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Díaz, J.I., Faghloumi, C. Actions on Environment under uncertainty: stochastic formulation and the associated deterministic problem. Rev. R. Acad. Cien. Serie A. Mat. 102, 335–353 (2008). https://doi.org/10.1007/BF03191827
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF03191827