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Composition and invariance methods for solving some stochastic control problems

Published online by Cambridge University Press:  01 July 2016

V. E. Beneš
V. E. Beneš*
Affiliation:
Bell Laboratories, Murray Hill, New Jersey
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Abstract

This paper considers certain stochastic control problems in which control affects the criterion through the process trajectory. Special analytical methods are developed to solve such problems for certain dynamical systems forced by white noise. It is found that some control problems hitherto approachable only through laborious numerical treatment of the non-linear Bellman-Hamilton-Jacobi partial differential equation can now be solved.

Keywords

STOCHASTIC CONTROLBELLMAN'S EQUATIONMAXIMUM PRINCIPLEGIRSANOV SOLUTIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 7 , Issue 2 , June 1975 , pp. 299 - 329
Copyright
Copyright © Applied Probability Trust 1975 

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References

[1] Girsanov, I. V. (1960) On transforming a certain class of stochastic processes by absolutely continuous substitution of measures. Theor. Probability Appl. 5, 285–301.CrossRefGoogle Scholar
[2] Wonham, W. H. (1968) On the separation theorem of stochastic control. SICON 6, 312–326.Google Scholar
[3] Beneš, V. E. (1974) Girsanov functionals, and optimal bang-bang laws for final value stochastic control. Stoch. Proc. Appl. 2, 127–140.Google Scholar
[4] Beneš, V. E. (1974) Full ‘bang’ to reduce predicted miss is optimal. To appear.Google Scholar
[5] Davis, M. H. A. and Varaiya, P. P. (1973) Dynamic programming conditions for partially observable stochastic systems. SICON 11, 226–261.Google Scholar
[6] Cameron, R. H. and Martin, W. T. (1944) Transformations of Wiener intergrals under translations. Ann. of Math. 45, 386–396.Google Scholar
[7] Beneš, V. E. (1971) Existence of optimal stochastic control laws. SICON 9, 446–472.Google Scholar
[8] Duncan, T. E. and Varaiya, P. P. (1971) On the solutions of a stochastic control system. SICON 9, 354–371.Google Scholar
[9] Friedman, A. (1964) Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs.Google Scholar
[10] Bellman, R. (1960) Introduction to Matrix Analysis. McGraw-Hill, New York.Google Scholar
[11] Freidlin, M. (1967) Quasilinear parabolic equations and measure in function space (in Russian). Funktsional. Anal. i Prilozhen. 1, 74–82.Google Scholar
[12] Nisio, M. (1973) Private communication of unpublished MS.Google Scholar
[13] McKean, H. P. Jr., (1969) Stochastic Integrals. Academic Press, New York and London.Google Scholar
[14] Beneš, V. E. and Shepp, L. A. (1968) Wiener integrals associated with diffusion proceses. Theor. Probability Appl. 13, 475–478.CrossRefGoogle Scholar
[15] Shepp, L. A. Jr. Klauder, J. R. and Ezawa, H. (1974) On the divergence of certain integrals of the Wiener process. To appear.Google Scholar
[16] Shepp, L. A. (1973) Private communication of unpublished MS.Google Scholar
[17] Kushner, H. (1963) Sufficient conditions for the optimality of a stochastic control. SICON 3, 507.Google Scholar