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Martingale methods in stochastic control

  • M. H. A. Davis
Part I: Survey Lectures
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 16)

Keywords

Stochastic Differential Equation Stochastic Control Minimum Principle Jump Process Bellman Equation 
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9. References

  1. [1]
    V.E. Beneš, Existence of optimal strategies based on specified information, for a class of stochastic decision problems, —SICON 8 (1970) 179–188Google Scholar
  2. [2]
    V.E. Beneš, Existence of optimal stochastic control laws, —SICON 9 (1971) 446–475Google Scholar
  3. [3]
    V.E. Beneš, Full "bang" to reduce predicted miss is optimal, —SICON 15 (1976) 52–83Google Scholar
  4. [4]
    A. Bensoussan and J.L. Lions, Applications des inéquations varationelles en contrôle stochastique, Dunod, Paris, 1978Google Scholar
  5. [5]
    J.M. Bismut, Théorie probabiliste du contrôle des diffusions, Mem. Americ. Math. Soc. 4 (1976), no. 167Google Scholar
  6. [6]
    J.M. Bismut, Duality methods in the control of densities, —SICON 16 (1978) 771–777Google Scholar
  7. [7]
    J.M. Bismut, An introductory approach to duality in optimal stochastic control, SIAM Review 20 (1978) 62–78Google Scholar
  8. [8]
    J.M. Bismut and B. Skalli, Temps d'arrêt optimal, théorie général des processus et processus de Markov, —ZW 39 (1977) 301–313Google Scholar
  9. [9]
    R. Boel and M. Kohlmann, Stochastic control over double martingales, in "Analysis and Optimization of Stochastic Systems" ed. O.L.R. Jacobs, Academic Press, New York/London 1979Google Scholar
  10. [10]
    R. Boel and M. Kohlmann, Semimartingale models of stochastic optimal control with applications to double martingales, preprint, Institut für Angewandte Mathematik der Universität Bonn, 1977Google Scholar
  11. [11]
    R. Boel and P. Varaiya, Optimal control of jump processes, —SICON 15 (1977) 92–119Google Scholar
  12. [12]
    R. Boel, P. Varaiya and E. Wong, Martingales on jump processes I and II, —SICON 13 (1975) 999–1061Google Scholar
  13. [13]
    P. Brémaud and J. Jacod, Processus ponctuels et martingales: resultats recents sur la modelisation et le filtrage, Adv. Appl. Prob. 9 (1977) 362–416Google Scholar
  14. [14]
    P. Brémaud and J.M. Pietri, The role of martingale theory in continuous-time dynamic programming, Tech. report, IRIA, Le Chesnay, France, 1978Google Scholar
  15. [15]
    J.M.C. Clark, The representation of functionals of Brownian motion by stochastic integrals, Ann. Math. Stat. 41 (1970) 1285–1295Google Scholar
  16. [16]
    M.H.A. Davis, On the existence of optimal policies in stochastic control, —SICON 11 (1973) 507–594Google Scholar
  17. [17]
    M.H.A. Davis, The representation of martingales of jump processes, —SICON 14 (1976) 623–638Google Scholar
  18. [18]
    M.H.A. Davis, The separation principle in stochastic control via Girsanov solutions, —SICON 14 (1976) 176–188Google Scholar
  19. [19]
    M.H.A. Davis, Functionals of diffusion processes as stochastic integrals, submitted to Math. Proc. Camb. Phil. Soc.Google Scholar
  20. [20]
    M.H.A. Davis, Nonlinear semigroups in the control of partially-observable stochastic systems, in Measure Theory and Applications to Stochastic Analysis, ed. G. Kallianpur and D. Kölzow, Lecture Notes in Mathematics, Springer-Verlag, to appearGoogle Scholar
  21. [21]
    M.H.A. Davis and J.M.C. Clark, "Predicted Miss" problems in stochatic control, Stochastics 2 (1979)Google Scholar
  22. [22]
    M.H.A. Davis and R.J. Elliott, Optimal control of a jump process, —ZW 40 (1977) 183–202Google Scholar
  23. [23]
    M.H.A. Davis and M. Kohlmann, Stochastic control by measure transformation: a general existence result, preprint, Institut für Angewandte Mathematik der Universität Bonn (1978)Google Scholar
  24. [24]
    M.H.A. Davis and P.P. Varaiya, Information states for linear stochastic systems, J. Math. Anal. Appl. 37 (1972) 387–402Google Scholar
  25. [25]
    M.H.A. Davis and P.P. Varaiya, Dynamic programming conditions for partially-observable stochastic systems, —SICON 11 (1973) 226–261Google Scholar
  26. [26]
    M.H.A. Davis and C.B. Wan, The principle of optimality for Markov jump processes, in "Analysis and Optimization of Stochastic Systems" ed. O.L.R. Jacobs, Academic Press, New York/London, 1979Google Scholar
  27. [27]
    C. Dellacherie, Capacités et processus stochastiques, Springer-Verlag, Berlin, 1972Google Scholar
  28. [28]
    C. Doléans-Dade, Quelques applications de la formule de changement de variables pour les semimartingales, —ZW 16 (1970) 181–194Google Scholar
  29. [29]
    T.E. Duncan, Dynamic programming criteria for stochastic systems in Riemannian manifolds, Applied Math. & Opt. 3 (1977) 191–208Google Scholar
  30. [30]
    T.E. Duncan and P.P. Varaiya, On the solutions of a stochastic control system, —SICON 9 (1971) 354–371Google Scholar
  31. [31]
    T.E. Duncan and P.P. Varaiya, On the solutions of a stochastic control system II, —SICON 13 (1975) 1077–1092Google Scholar
  32. [32]
    R.J. Elliott, The existence of value in stochastic differential games, —SICON 14 (1976) 85–94Google Scholar
  33. [33]
    R.J. Elliott, Double martingales, —ZW 34 (1976) 17–28Google Scholar
  34. [34]
    R.J. Elliott, The optimal control of a stochastic system, —SICON 15 (1977) 756–778Google Scholar
  35. [35]
    R.J. Elliott, The existence of optimal strategies and saddle points in stochastic differential games, in Differential Games and Applications, ed. P. Hagedorn, Lecture Notes in Control and Information Sciences 3, Springer-Verlag, Berlin, 1977Google Scholar
  36. [36]
    R.J. Elliott, Lévy systems and absolutely continuous changes of measure, J. Math. Anal. Appl. 61 (1977) 785–796Google Scholar
  37. [37]
    R.J. Elliott, The optimal control of a semimartingale, 3rd Kingston Conference on Control Theory, Kingston, RI (1978)Google Scholar
  38. [38]
    R.J. Elliott, The martingale calculus and its applications, this volumeGoogle Scholar
  39. [39]
    R.J. Elliott and P.P. Varaiya, A sufficient condition for the optimal control of a partially observed stochastic system, in "Analysis and Optimization of Stochastic Systems," ed. O.L.R. Jacobs, Academic Press, New York/London, 1979Google Scholar
  40. [40]
    W.H. Fleming, Optimal continuous-parameter stochastic control, SIAM Rev. 11 (1969) 470–509Google Scholar
  41. [41]
    W.H. Fleming and R.W. Rishel, Deterministic and stochastic optimal control, Springer-Verlag, New York, 1975Google Scholar
  42. [42]
    M. Fujisaki, G. Kallianpur and H. Kunita, Stochastic differential equations for the nonlinear filtering problem, Osaka J. Math. 9 (1972) 19–40Google Scholar
  43. [43]
    I.V. Girsanov, On transforming a certain class of stochastic processes by absolutely continuous substitution of measures, Theory of Prob. and Appls. 5, (1960) 285–301Google Scholar
  44. [44]
    U.G. Haussmann, On the stochastic maximum principle, —SICON 16 (1978) 236–251Google Scholar
  45. [45]
    U.G. Haussmann, Functionals of Ito processes as stochastic integrals, —SICON 16 (1978) 252–269Google Scholar
  46. [46]
    U.G. Haussmann, On the integral representation of functionals of Ito processes, Stochastic 2 (1979)Google Scholar
  47. [47]
    J. Jacod, Multivariate point processes: predictable projection, Radon-Nikodym derivatives, representation of martingales, —ZW 31 (1975) 235–253Google Scholar
  48. [48]
    J. Jacod and J. Memin, Caracteristiques locales et conditions de continuité absolue pour les semimartingales, —ZW 35 (1976) 1–37Google Scholar
  49. [49]
    M. Kohlmann, A game with Wiener noise and jump process disturbances, submitted to StochasticsGoogle Scholar
  50. [50]
    M. Kohlmann, On control of jump process, preprint, Institut für Angewandte Mathematik der Universität Bonn, 1978Google Scholar
  51. [51]
    N.V. Krylov, Control of a solution of a stochastic integral equation Theory of Prob. and Appls. 17 (1972) 114–131Google Scholar
  52. [52]
    H. Kunita and S. Watanabe, On square-integrable martingales, Nagoya Math. Journal 30 (1967) 209–245Google Scholar
  53. [53]
    H.J. Kushner, Necessary conditions for continuous-parameter stochastic optimization problems, —SICON 10 (1972) 550–565Google Scholar
  54. [54]
    H.J. Kushner, Probability Methods for Approximations in Stochastic Control and for Elliptic Equations, Academic Press, New York, 1977Google Scholar
  55. [55]
    H.J. Kushner, Optimality conditions for the average cost per unit time problem with a diffusion model, —SICON 16 (1978) 330–346Google Scholar
  56. [56]
    E. Lenglart, Transformation des martingales locales par changement absolument continu de probabilités, —ZW 39 (1977) 65–70Google Scholar
  57. [57]
    D. Lepingle and J. Memin, Sur l'integrabilité uniforme des martingales exponentielles, —ZW 42 (1978) 175–203Google Scholar
  58. [58]
    J.P. Lepeltier and B. Marchal, Sur l'existence de politiques optimales dans le contrôle integro-differentiel, Ann. Inst. H. Poincaré 13 (1977) 45–97Google Scholar
  59. [59]
    J.P. Lepeltier and B. Marchal, Techniques probabilistes dans le contrôle impulsionelle, Stochastics 2 (1979)Google Scholar
  60. [60]
    R.S. Liptser and A.N. Shiryayev, Statistics of Random Processes, vols I and II, Springer-Verlag, New York, 1977Google Scholar
  61. [61]
    J. Memin, Conditions d'optimalité pour un problème de contrôle portant sur une famille de probabilités dominées par une probabilité P, preprint, University of Rennes, 1977Google Scholar
  62. [62]
    J.F. Mertens, Processus stochastiques généraux, applications aux surmartingales, —ZW 22 (1972) 45–68Google Scholar
  63. [63]
    P.A. Meyer, Probability and Potentials, Blaisdell, Waltham, MA, 1966Google Scholar
  64. [64]
    P.A. Meyer, Un cours sur les intégrales stochastiques, —SP10/LNMGoogle Scholar
  65. [65]
    R.E. Mortensen, Stochastic optimal control with noisy observations, Int. J. Control 4 (1966) 455–464Google Scholar
  66. [66]
    M. Nisio, On a nonlinear semigroup attached to stochastic control, Pub. Res. Inst. Math. Sci., Kyoto, 12 (1976) 513–537Google Scholar
  67. [67]
    A.A. Novikov, On an identity for stochastic integrals, Theor. Probability Appl. 17 (1972) 717–720Google Scholar
  68. [68]
    R. Rishel, Necessary and sufficient conditions for continuous-time stochastic optimal control, —SICON 8 (1970) 559–571Google Scholar
  69. [69]
    J.H. van Schuppen and E. Wong, Transformations of local martingales under a change of law, Ann. Prob. 2 (1974) 879–888Google Scholar
  70. [70]
    C. Striebel, Martingale conditions for the optimal control of continuous-time stochastic systems, 5th Symposium on nonlinear estimation and applications, San Diego, CA, 1974Google Scholar
  71. [71]
    D.W. Stroock and S.R.S. Varadhan, Diffusion processes with continuous coefficients, Comm. Pure and Appl. Math. 22 (1969) 345–400, 479–530Google Scholar
  72. [72]
    B.S. Tsyrelson, An example of a stochastic equation having no strong solution, Theory of Prob. and Appls. 20 (1975) 427–430Google Scholar
  73. [73]
    K. Uchida, On the existence of a Nash equilibrium point in N-person nonzero sum stochastic differential games, —SICON 16 (1978) 142–149Google Scholar
  74. [74]
    K. Uchida, A note on the existence of a Nash equlibrium point in stochastic differential games, to appear in SICON SIAM Journal on Control (and Optimization)Google Scholar
  75. [75]
    P. Varaiya, Differential games, Proc. VI Berkeley Symposium on Math. Stat. and Prob., vol. 3, 687–697, Univ. of California Press, Berkeley, 1972Google Scholar
  76. [76]
    P.P. Varaiya, N-person stochastic differential games, —SICON 14 (1976) 538–545Google Scholar
  77. [77]
    C.B. Wan and M.H.A. Davis, existence of optimal controls for stochastic jump processes, to appear in SICON SIAM Journal on Control (and Optimization)Google Scholar
  78. [78]
    W.M. Wonham, On the separation theorem of stochastic control, —SICON 6 (1968) 312–326Google Scholar
  79. [79]
    C. Yoeurp, Decompositions des martingales locales et formules exponentielles, —SP10/LNM 511 (1976) 432–480Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • M. H. A. Davis
    • 1
  1. 1.Laboratory for Information and Decision SystemsMassachusetts Institute of TechnologyCambridge

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