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Viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions. Part II: Optimal control of Zakai's equation

Part of the Lecture Notes in Mathematics book series (LNM,volume 1390)

Abstract

We consider here the value function corresponding to the optimal control of Zakai's equation and we study various regularity properties of this value function in the context of L2 distributions. In particular, we show that it is the unique viscosity solution of the corresponding infinite dimensionnal Hamilton-Jacobi-Bellman equation.

Key-words

  • Viscosity solutions
  • dynamic programming
  • optimal stochastic control
  • Hamilton-Jacobi-Bellman equations
  • partial observations
  • Zakai's equation
  • fully nonlinear second-order equations
  • infinite dimensions
  • degenerate ellipticity

Work partially supported by US Army contract DASA 45-88-C-009.

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Bibliography

  1. P.L. Lions. Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part I: The case of bounded stochastic evolutions. Preprint.

    Google Scholar 

  2. M.G. Crandall and P.L. Lions. Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc., 277 (1983), p. 1–42; see also C.R. Acad. Sci. Paris, 292 (1981), p. 183–186.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. M.G. Crandall and P.L. Lions. Hamilton-Jacobi equations in infinite dimensions. J. Funct. Anal., Part I, 62 (1985), p. 379–396; Part II, 65 (1986), p. 368–405; Part III, 68 (1986), p. 214–247.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. M.G. Crandall, L.C. Evans and P.L. Lions. Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc., 282 (1984), p. 487–502.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. P.L. Lions. Generalized solutions of Hamilton-Jacobi equations. Pitman, London, 1982.

    MATH  Google Scholar 

  6. P.L. Lions. Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 2. Comm. P.D.E., 8 (1983), p. 1229–1276.

    CrossRef  MATH  Google Scholar 

  7. W.H. Fleming and R. Rishel. Deterministic and stochastic optimal control. Springer, Berlin, 1975.

    CrossRef  MATH  Google Scholar 

  8. A. Bensoussan. Stochastic control by functional analysis methods. North-Holland, Amsterdam, 1982.

    MATH  Google Scholar 

  9. N.V. Krylov. Controlled diffusion processes. Springer, Berlin, 1980.

    CrossRef  MATH  Google Scholar 

  10. P.L. Lions. On the Hamilton-Jacobi-Bellman equations. Acta Applic., 1 (1983), p. 17–41.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. W.H. Fleming and E. Pardoux. Optimal control for partially observed diffusions. SIAM J. Control Optim., 20 (1982), p. 261–285.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. J.M. Bismut. Partially observed diffusions and their control. SIAM J. Control Optim., 20 (1982), p. 302–309.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. V.E. Benes and I. Karatzas. On the relation of Zakai's and Mortensen's equations. SIAM J. Control Optim., 21 (1983), p. 472–489.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. U.G. Haussmann. On the existence of optimal controls for partially observed diffusions. SIAM J. Control Optim., 20 (1982), p. 385–407.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. V.S. Borkar. Existence of optimal controls for partially observed diffusions. Stochastics, 11 (1983), p. 103–142.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. A. Bensoussan. Maximum principle and dynamic programming approaches of the optimal control of partially observed diffusions. Stochastics, 9 (1983), p. 169–222.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. E. Pardoux. Stochastic partial differential equations and filtering of diffusion processes. Stochastics, 3 (1979), p. 127–167.

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. R.J. Elliott and M. Kohlmann. On the existence of optimal controls for partially observed diffusions. Appl. Math. Optim., 9 (1982), p. 41–67.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. M. Kohlmann. Existence of optimal controls for a partially observed semi-martingale control problem. Stoch. Proc. Appl., 13 (1982), p. 215–226.

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. M. Kohlmann. A maximum principle and dynamic programming for a partially observed control problem. Preprint 86, Univ. of Konstanz, 1985.

    Google Scholar 

  21. N. Cutland. Optimal controls for partially observed stochastic systems using nonstandard analysis. Lecture Notes Control Inf. Sc. no43, Springer, 1982, p. 276–284.

    Google Scholar 

  22. N. Christopeit. Existence of optimal controls under partial observation. Z. Wahrs. verw. Geb., 51 (1980), p. 201–213.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. N. Christopheit and M. Kohlmann. Some recent results on the control of partially observable stochastic systems. Lecture Notes Control Inf. Sci. no43, Springer, 1982, p. 251–275.

    Google Scholar 

  24. Y. Fujita. Nonlinear semigroup for the unnormalized conditional density. Tôhoku Math. J., 37 (1985), p. 251–263.

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. Y. Fujita and M. Nisio. Nonlinear semigroups associated with optimal stopping of controlled diffusions under partial observation. Comp. Math. Appl., 12 (1986), p. 749–760.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. H. Kunita. Stochastic partial differential equations connected with nonlinear filtering. Proc. CIME School in Nonlinear Filtering and Stochastic Control, 1982.

    Google Scholar 

  27. G. Kallianpur. Stochastic filtering theory. Springer, Berlin, 1980.

    CrossRef  MATH  Google Scholar 

  28. S.K. Mitter. Nonlinear filtering of diffusion process, a guided tour. Lecture Notes Control Inf. Sc. no42, Springer, 1982, p. 256–266.

    Google Scholar 

  29. Q. Zhang. Controlled partially observed diffusions. Preprint.

    Google Scholar 

  30. W.H. Fleming. Nonlinear semigroup for controlled partially observed diffusions. SIAM J. Control Optim., 20 (1982), p. 261–285.

    CrossRef  MathSciNet  MATH  Google Scholar 

  31. G. Da Prato. Some results on Bellman equations in Hilbert spaces. SIAM J. Control Optim., 23 (1985), p. 61–71.

    CrossRef  MathSciNet  MATH  Google Scholar 

  32. G. Da Prato. Some results on Bellman equations in Hilbert spaces and applications to infinite dimensional control problems. Lecture Notes Control Inf. Sc. no69, Springer, Berlin, 1985.

    MATH  Google Scholar 

  33. M.H.A. Davis. Nonlinear semigroups in the control of partially observable stochastic systems. In Measure Theory and Applications to Stochastic Analysis, Lecture Notes Math. no695, Springer, Berlin, 1975.

    Google Scholar 

  34. M.G. Crandall and P.L. Lions. Hamilton-Jacobi equations in infinite dimensions. Part IV. Hamiltonians with unbounded linear terms. Preprint. See also C.R. Acad. Sci. Paris, 305 (1987), p. 233–236.

    MathSciNet  Google Scholar 

  35. P.L. Lions. Some recent results in the optimal control of diffusion processes. In Stochastic Analysis, Proceedings of the Taniguchi International Symposium on Stochastic Analysis, Katata and Kyoto, 1982. Kinukiniya, Tokyo, 1984.

    MATH  Google Scholar 

  36. R. Jensen. The maximum principle for viscosity solutions of fully nonlinear second-order partial differential equations. Preprint.

    Google Scholar 

  37. R. Jensen, P.L. Lions and P.E. Souganidis. A uniqueness result for viscosity solutions of second-order fully nonlinear partial differential equations. Proc. A.M.S., 1988.

    Google Scholar 

  38. P.L. Lions and P.E. Souganidis. Viscosity solutions of second-order equations, stochastic control and stochastic differential games. In Stochastic Differential Systems, Stochastic Control Theory and Applications. Eds. W.H. Fleming and P.L. Lions, IMA Volume 10, Springer, Berlin, 1988.

    Google Scholar 

  39. H. Ishii. On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDE's. Preprint.

    Google Scholar 

  40. H. Ishii and P.L. Lions. Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. Preprint.

    Google Scholar 

  41. R. Jensen. Uniqueness criteria for viscosity solutions of fully nonlinear elliptic partial differential equations. Preprint.

    Google Scholar 

  42. P.L. Lions. Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part III. Preprint.

    Google Scholar 

  43. I. Ekeland. Nonconvex minimization problems. Bull. Amer. Math. Soc., 1 (1979), p. 443–474.

    CrossRef  MathSciNet  MATH  Google Scholar 

  44. C. Stegall. Optimization of functions on certain subsets of Banach spaces. Math. Annal., 236 (1978), p. 171–176.

    CrossRef  MathSciNet  MATH  Google Scholar 

  45. J. Bourgain. La propriété de Radon-Nikodym. Cours de 3e cycle no36, Univ. P. et M. Curie, Paris, 1979.

    MATH  Google Scholar 

  46. J.M. Lasry and P.L. Lions. A remark on regularization in Hilbert spaces. Isr. J. Math., 55 (1986), p. 145–147.

    CrossRef  MathSciNet  MATH  Google Scholar 

  47. O. Hijab. Partially observed control of Markov processes. I,II,III. Preprint.

    Google Scholar 

  48. N. El Karoui, D. Huu Nguyen and M. Jeanblanc-Piqué. Existence of an optimal markovian control for the control under partial observations. Preprint.

    Google Scholar 

  49. W.H. Fleming and M. Nisio. On stochastic relaxed controls for partially observed diffusions. Nagoya Math. J., 93 (1984), p. 71–108.

    MathSciNet  MATH  Google Scholar 

  50. T.G. Kurtz and D.L. Ocone. A martingale problem for conditional distributions and uniqueness for the nonlinear filtering equations. Ann. Proba., 16 (1988), p. 80–107.

    CrossRef  MathSciNet  MATH  Google Scholar 

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Lions, P.L. (1989). Viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions. Part II: Optimal control of Zakai's equation. In: Da Prato, G., Tubaro, L. (eds) Stochastic Partial Differential Equations and Applications II. Lecture Notes in Mathematics, vol 1390. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083943

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  • DOI: https://doi.org/10.1007/BFb0083943

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