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Mixed control problem under partial observation

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Abstract

In this paper, we consider the control problem with optimal stopping of a jump process. Using compactification methods, we obtain the existence of an optimal Markovian optimal control.

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References

  1. P. Billingsley, Convergence of Probability Measures, Addison-Wesley, Reading, MA, 1969.

    Google Scholar 

  2. J.-M. Bismut, Un problème de contrôle stochastique avec observation partielle, Z. Wahrsch. Verw. Gebiete, 49 (1979), pp. 63–95.

    Google Scholar 

  3. V. Borkar, Optimal Control of Diffusion Processes. Pitman Research Notes in Mathematics, Vol. 203, Pitman, London, 1989.

    Google Scholar 

  4. P. Brémaud, Point Processes, and Queues, Springer-Verlag, New York, 1981.

    Google Scholar 

  5. C. Dellacherie and P. A. Meyer, Probabilités et Potentiel, Chapitres I–IV, Hermann, Paris, 1975.

    Google Scholar 

  6. N. El Karoui, D. Huu Nguyen, and M. Jeanblanc-Picqué, Existence of an optimal Markovian filter for the control under partial observations, SIAM J. Control Optim., 26, No. 5 (1988), pp. 1025–1061.

    Google Scholar 

  7. N. El Karoui, D. Huu Nguyen, and M. Jeanblanc-Picqué, Compactification methods in the control of degenerate diffusions: existence of an optimal control, Stochastics, 20 (1987), pp. 169–219.

    Google Scholar 

  8. N. El Karoui, Les aspects probabilistes du contrôle stochastique, in Ecole d'Été de Probabilités de Saint-Flour, Lecture Notes in Mathematics, Vol. 876, Springer-Verlag, New York, 1979, pp. 74–239.

    Google Scholar 

  9. N. El Karoui and I. Karatzas, Probabilistic aspects of finite-fuel, reflected follower problems, Stochastics (1989) (to appear).

  10. S. Ethier and T. Kurtz, Markov Processes, Characterization and Convergence, Wiley, New York, 1986.

    Google Scholar 

  11. N. El Karoui, J. P. Lepeltier, and B. Marchal, Optimal stopping of controlled Markov processes, in Advances in Filtrage and Optimal Stochastic Control, Proc. Cocoyoc 1982, Lecture Notes in Control and Information Sciences, Vol. 61, Springer-Verlag, Berlin, 1982, pp. 91–105.

    Google Scholar 

  12. N. El Karoui, J. P. Lepeltier, and A. Millet, A probabilistic approach to the reduite, Polish J. Math. (1988) (to appear).

  13. W. Fleming, Generalized solutions in optimal stochastic control, in Differential Games and Control Theory, Lecture Notes in Pure and Applied Mathematics, Vol. 30, Marcel Dekker, New York, 1978.

    Google Scholar 

  14. W. Fleming, Measure-valued processes in the control of partially-observable stochastic systems, Appl. Math. Optim., 6 (1980), pp. 271–285.

    Google Scholar 

  15. U. Haussmann, Existence of optimal markovian controls for degenerate diffusions, in Proceedings of the Third Bad-Honnef Conference, Lecture Notes in Control and Information Sciences, Vol. 78, Springer-Verlag, Berlin, 1986.

    Google Scholar 

  16. O. Hijab, Partially observed control of Markov processes, I, Stochastics (1990) (to appear).

  17. U. Haussman and J. P. Lepeltier, On the existence of optimal controls, SIAM J. Control Optim., 28, No. 4 (1990), pp. 851–902.

    Google Scholar 

  18. J. Jacod and J. Mémin, Existence of weak solutions for stochastic differential equations with driving semimartingales, Stochastics, 4 (1981), pp. 317–337.

    Google Scholar 

  19. A. Joffe and M. Métivier, Weak convergence of sequence of semimartingales with application to multitype branching process, Adv. in Appl. Probab. 18 (1986), pp. 20–65.

    Google Scholar 

  20. J. Jacod and A. Shiriyaev, Limit Theorems for Stochastic Processes, Springer-Verlag, New York, 1988.

    Google Scholar 

  21. K. Kishino, On the separation principle of stochastic control, J. of Tokyo Univ., 1982.

  22. T. Kurtz and D. Ocone, A martingale problem for conditional distributions and uniqueness for the nonlinear filtering equation, Lecture Notes in Control and Information Sciences, Vol. 69, Springer-Verlag, Berlin, 1984.

    Google Scholar 

  23. H. Kushner, Numerical methods for stochastic control problems in continuous time, SIAM J. Control Optim., 28, No. 4 (1990), pp. 999–1048.

    Google Scholar 

  24. N. V. Krylov, Controlled Diffusions Processes, Springer-Verlag, New York, 1974.

    Google Scholar 

  25. T. Kunita, Asymptotic behaviour of the non-linear filtering errors of Markov processes, J. Multivariate Anal., 1 (1971), pp. 365–393.

    Google Scholar 

  26. J. Jacod, Calcul stochastique et problèmes de martingales. Lecture Notes in Mathematics Vol. 714, Springer-Verlag, New York, 1979.

    Google Scholar 

  27. J. P. Lepeltier and B. Marchai, Problème des martingales et équations différentielles associées à un opérateur intégro-différentiel, Ann. Inst. H. Poincaré Sect. B, 12 (1976), pp. 43–103.

    Google Scholar 

  28. R. C. Lipster and A. N. Shiriyaev, Statistics of Random Processes, Springer-Verlag, New York, 1977.

    Google Scholar 

  29. L. Mazliak, Un problème de contrôle stochastique avec sauts, Note aux CRAS de Paris, Série I, Vol. 308 (1989).

  30. L. Mazliak, Problème de contrôle mixte avec sauts, Note aux CRAS de Paris, Série I, Vol. 309 (1989).

  31. S. Roeliy-Coppoletta, A criterion of convergence of measure valued processes: Application to measure-branching processes, Stochastics, 7 (1986), pp. 43–66.

    Google Scholar 

  32. A. N. Shiriyaev, Optimal Stopping Rules, Applications of Mathematics, Springer-Verlag, Berlin, 1977.

    Google Scholar 

  33. J. Szpirglas and R. Mazzioto, Modèle général de filtrage non-linéaire, et équations différentielles stochastiques associées, Ann. Inst. H. Poincaré Sect. B, 2 (1979), pp. 147–173.

    Google Scholar 

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Communicated by A. Bensoussan

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Mazliak, L. Mixed control problem under partial observation. Appl Math Optim 27, 57–84 (1993). https://doi.org/10.1007/BF01182598

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