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On stochastic relaxed control for partially observed diffusions

  • W. H. Fleming (a1) and M. Nisio (a2)
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In this paper we are concerned with stochastic relaxed control problems of the following kind. Let X(t), t ≥ 0, denote the state of a process being controlled, Y(t), t ≥ 0, the observation process and p(t, ·) a relaxed control, that is a process with values probability measures on the control region Г. The state and observation processes are governed by stochastic differential equations

and

where B and W are independent Brownian motions with values in Rn and Rm respectively, (put m = 1 for simplicity).

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COPYRIGHT: © Editorial Board of Nagoya Mathematical Journal 1984
References
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[ 1 ] Bensoussan, A. and Lions, J. L., Application des inéquations variationnelles en contrôle stochastique, Dunod, Paris 1978.
[ 2 ] Billingsley, P., Convergence of Probability Measures, John Wiley 1968.
[ 3 ] Bismut, J. M., Partially observed diffusions and their control, SIAM. J. Control Optim., 20 (1982), 302–309.
[ 4 ] Davis, M. H. A. and Kohlmann, M., On the nonlinear semigroup of stochastic control under partial observations, preprint.
[ 5 ] Fleming, W. H., Generalized solutions in optimal stochastic control, Differential games and control theory II, Lect. Notes in pure and Appl. math., 30, Dekker.
[ 6 ] Fleming, W. H., Nonlinear semigroup for controlled partially observed diffusions, SIAM. J. Control Optim., 20 (1982), 286–301.
[ 7 ] Fleming, W. H. and Nisio, M., On the existence of optimal stochastic controls, J. Math. Mech., 15 (1966), 777–794.
[ 8 ] Fleming, W. H. and Pardoux, E., Optimal control for partially observed diffusions, SIAM. J. Control Optim., 20 (1982), 258–285.
[ 9 ] Haussman, U. G., On the existence of optimal controls for partially observed diffusions, SIAM. J. Control Optim., 20 (1982), 385–407.
[10] Nisio, M., Stochastic Control Theory, ISI Lect. Note 9, McMillan India, 1981.
[11] Parthasarathy, K. R., Probability Measures on Metric Spaces, Acad. Press, 1967.
[12] Skorobod, A. V., Studies in the Theory of Random Processes, Addism-Wesly, 1965 (orig. Russian).
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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
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