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Abstract

The purpose of this lecture is to survey the main aspects of stochastic control theory and its applications. To try to be exhaustive is impossible, since the applications are extremely diversified from engineering to management, statistics,…, and the methods make use of many disciplines such as functional analysis, partial differential equations, probability theory and stochastic processes, control theory and optimization, game theory, numerical analysis among the main ones. From the results point of view there are basically three parts. Firstly the situation of one (or several) decision maker with complete information, where the theory is well understood and several numerical algorithms exist. Secondly, the situation of one decision maker with partial information, where the general theory is understood, but is difficult to apply. There, progress has to be made in the search of particular cases where the solution simplifies, or in the search of more efficient algorithms, making use for instance of parallel processing. Finally, the case of several decision makers with partial information is widely open on the theory side, although some heuristic algorithms are available to handle practical applications.

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References

  1. [1]
    A. Bensoussan, Stochastic Maximum Principle, Encyclopedia of Systems and Controls, ed. M. P. Singh, Pergamon Press, 1984.Google Scholar
  2. [2]
    A. Bensoussan-J. Lesourne, Growth of Firms: A Stochastic Control Theory Approach, in book edited in honor of Prof. Albach by K. Brockhoff & W. Krelle, Springer Verlag, 1981.Google Scholar
  3. [3]
    A. Bensoussan-J. L. Lions, Applications of Variational Inequalities in Stochastic Control, North Holland, 1982.Google Scholar
  4. [4]
    A. Bensoussan-J. Van Schuppen, Stochastic Control with Partial Information for an Exponential of an Integral Payoff, SIAM Control, 1984.Google Scholar
  5. [5]
    D. P. Bertsekas, Dynamic Programming and Stochastic Control, Academic Press, 1976.Google Scholar
  6. [6]
    J.M. Bismut, An introducing Approach to Duality in Optimal Stochastic Control, SIAM Review, Vol. 20, n°l, Jan. 1978.Google Scholar
  7. [7]
    A. Breton-C. Leguay, Application du contrôle stochastique à la gestion des centrales thermiques et hydrauliques; Lecture Notes Econ. Math. Systems, 107, Springer, 1975, pp. 728–744.CrossRefGoogle Scholar
  8. [8]
    P. Colleter-F. A. Delebecque-F. Falgarone-J. P. Quadrat, Application of Stochastic Control Methods to the Management of Energy Production in New Caledonia, in Applied Stochastic Control in Econometrics and Management Science, ed. by A. Bensoussan, P. Kleindorfer & C. Tapiero, North-Holland, 1980.Google Scholar
  9. [9]
    F. Delebecque-J. P. Quadrat, Contribution of Stochastic Control, Singular Perturbation, Averaging and Team Theories to an Example of Large Scale Systems, IEEE Special Issue on Large Scale Systems.Google Scholar
  10. [10]
    W. Fleming-R. Rishel, Optimal Deterministic and Stochastic Control, Springer Verlag, Berlin (1975).CrossRefGoogle Scholar
  11. [11]
    A. Friedman, Stochastic Differential Equations, Vol. I, II, Academic Press, N. Y., 1976.Google Scholar
  12. [12]
    N. Krylov, Controlled Diffusion Processes, Springer Verlag, Berlin, 1980.CrossRefGoogle Scholar
  13. [13]
    H. J. Kushner, Necessary Conditions for Continuous Parameter Stochastic Optimization Problems, SIAM J. Cont., 10, (1972), pp. 550–565.Google Scholar
  14. [14]
    C. Leguay, Applications of Stochastic Control to the Problem of Optimal Energy Management, in Applied Stochastic Control in Econometrics and Management, loc. cit.Google Scholar
  15. [15]
    P. L. Lions, Optimal Control of Diffusion Processes and Hamilton-Jacobi-Bellman Equations, Part I, II, Comm. P.D.E., 8, (1983).Google Scholar
  16. [16]
    R. C. Merton, Theory of Rational Option Pricing, Bell. J. Eco. Magt. Science, Vol. 4, pp. 141–183, (1973).CrossRefGoogle Scholar
  17. [17]
    H. Scarf, The Optimality of (S, s) Policies in the Dynamic Inventory Problem, in K.J. Arrow, S. Karlin & P. Suppes eds, Mathematical Methods in the Social Sciences, Stanford University Press, (1960).Google Scholar
  18. [18]
    A.N. Shiryaev, Sequential Statistical Analysis, AMS Pub. Providence, (1973).Google Scholar
  19. [19]
    P. Whittle, Optimization over Time, Vol. I, II, J. Wiley, (1983).Google Scholar
  20. [20]
    W.M. Wonham, On the Separation Principle of Stochastic Control, SIAM Cont, (1968), vol. 6, n°2.Google Scholar

Copyright information

© Springer Fachmedien Wiesbaden 1985

Authors and Affiliations

  • Alain Bensoussan
    • 1
  1. 1.ParisFrance

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