×
Chitashvili, R. J.; Mania, M. G.

Optimal locally absolutely continuous change of measure. Finite set of decisions. II: Optimization problems. (English) Zbl 0639.93065

Stochastics 21, 187-229 (1987).
This paper continues part I [the authors, ibid. 21, 131-185 (1987; Zbl 0622.93077)]. The authors consider control problems, where a control choice implies a locally absolutely continuous transformation of some initial measure. They investigate a martingale characterization of the value process and derive a nonlinear stochastic differential equation which is a nonlinear analogue of the inverse Girsanov transformation standing for the Bellman equation. Using these characterizations, the defect formula, the maximum principle, the extremal set of controls and sufficient filtration for the optimization problem are considered.
Reviewer: M.Nisio

Cited in 1 Review
Cited in 2 Documents

MSC:

93E20 Optimal stochastic control
49K45 Optimality conditions for problems involving randomness
49L20 Dynamic programming in optimal control and differential games
60G44 Martingales with continuous parameter
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

nonlinear stochastic differential equation; inverse Girsanov transformation; Bellman equation; defect formula; maximum principle; extremal set of controls; sufficient filtration

Citations:

Zbl 0622.93077
Cite Review PDF
Full Text: DOI

References:

[1] Chitashvili R. J., Martingale ideology in the theory of controlled stochastic processes Lecture Notes in Mathematics, No. 1021 pp 73– (1983) · Zbl 0543.93071
[2] DOI: 10.1137/0309034 · doi:10.1137/0309034
[3] DOI: 10.1137/0308040 · Zbl 0206.45804 · doi:10.1137/0308040
[4] DOI: 10.1137/0311020 · Zbl 0258.93029 · doi:10.1137/0311020
[5] Striebel C., Workshop on Stoch. Filtering and Control (1974)
[6] Elliot R. J., Stochastic Calculus and Applications (1982) · JFM 14.0416.01
[7] El Karoui N., Ecole d’Eté de St Flour (1980)
[8] Davis M. H. A., Lecture Notes in Control and Information Sciences (1979)
[9] Gamkrelidze R. V., Doklady Akademii Nauk 6 pp 1243– (1962)
[10] Fleming W.H., Nagoya Math. J 93 pp 71– (1984) · Zbl 0506.60055 · doi:10.1017/S0027763000020742
[11] Kabanov Yu. M., Mathematicheski Sbornik 119, No. 191 11 (1982)
[12] Yushkevich A. A., Russian Mathem. Surveys 37 6 pp 241– (1982)
[13] Jacod J., Lecture Notes in Mathematics pp 714–
[14] DOI: 10.2307/1402605 · Zbl 0478.60062 · doi:10.2307/1402605
[15] Yu. M. Kabanov, Mathemati- chesky Sbornik 107 3 pp 364, 32– (1978)
[16] Neveu J., Bases Mathematiques du Calcul des Probabilites (1964) · Zbl 0175.16601
[17] Dellacherie C., Ergeb. der Math, und Grenzg (1972)
[18] Chitashvili R. J., stochastics 4 pp 281– (1981) · Zbl 0454.60056 · doi:10.1080/17442508108833168
[19] Yosida K., Functional Analysis (1965) · Zbl 0126.11504
[20] Fleming W. H., J. Math. Mech 15 pp 794– (777)
[21] DOI: 10.1137/0311045 · Zbl 0238.93044 · doi:10.1137/0311045
[22] Shiryayev A. N., Statistics of Random Processes (1978) · Zbl 0369.60001
[23] Krylov N. V., Controlled Processes of Diffusion Type (1977) · Zbl 0513.60043
[24] Chitashvili, R. J. and Mania, M. G. 1986. ”On optimal controls in the problem of locally absolutely continuous change of measure. Proceedings of the IV Int”. VNU Science Press. · Zbl 0652.60051
[25] DOI: 10.1137/0320021 · Zbl 0484.93077 · doi:10.1137/0320021
[26] Pressman E. L., On the control of multidimensional counting process depending on unknown parameter. IV, Int. Vilnius Conf. on Probability Theory and Math pp 53– (1985)
[27] DOI: 10.1137/0315048 · Zbl 0359.93046 · doi:10.1137/0315048
[28] DOI: 10.1137/0316015 · Zbl 0373.93053 · doi:10.1137/0316015
[29] Chirashvili R. J., Martingale approach to partially observable controlled stochastic systems (1984)
[30] Kushner H.J., SI AM J. Control 2 pp 587– (1973)
[31] DOI: 10.1137/0316015 · Zbl 0373.93053 · doi:10.1137/0316015
[32] Kabanov Yu. M., Prob. Theory and Related Fields
[33] DOI: 10.1070/RM1972v027n03ABEH001378 · Zbl 0256.49025 · doi:10.1070/RM1972v027n03ABEH001378
[34] Memin, J. 1978. ”Decompositions multiplicatives de semimartingales exponentielles et applications, Lecture Notes in Mathematics”. Edited by: Memin, J. 36–45. Springer-Verlag. · Zbl 0384.60034
[35] Chitashvili R. J., Stochastics 21 pp 131– (1987) · Zbl 0622.93077 · doi:10.1080/17442508708833455
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.