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Optimality conditions of controlled backward doubly stochastic differential equations. (English) Zbl 1226.93136

Summary: We introduce and study the optimality conditions for stochastic control problems of nonlinear backward doubly stochastic differential equations. Necessary and sufficient optimality conditions, where the control domain is convex and the coefficients depend explicitly on the variable control, are proved. The results are stated in the form of weak stochastic maximum principle, and under additional hypotheses, we give these results in the global form. This is the first version of the stochastic maximum principle that covers the backward-doubly systems.

MSC:

93E20 Optimal stochastic control
49K45 Optimality conditions for problems involving randomness
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Arkin V., Soviet Math. Dokl. 20 pp 1– (1979)
[2] DOI: 10.1137/070681053 · Zbl 1167.49024 · doi:10.1137/070681053
[3] DOI: 10.1515/156939706778239846 · Zbl 1113.93105 · doi:10.1515/156939706778239846
[4] Bensoussan A., Lecture Notes in Mathematics 972 pp 40– (1982)
[5] DOI: 10.1016/0022-247X(73)90066-8 · Zbl 0276.93060 · doi:10.1016/0022-247X(73)90066-8
[6] DOI: 10.1137/0314028 · Zbl 0331.93086 · doi:10.1137/0314028
[7] DOI: 10.1137/1020004 · Zbl 0378.93049 · doi:10.1137/1020004
[8] DOI: 10.1137/S0363012992240722 · Zbl 0826.93069 · doi:10.1137/S0363012992240722
[9] DOI: 10.1006/jmaa.1999.6515 · Zbl 0937.93053 · doi:10.1006/jmaa.1999.6515
[10] DOI: 10.1111/1467-9965.00022 · Zbl 0884.90035 · doi:10.1111/1467-9965.00022
[11] DOI: 10.1007/BF01447329 · Zbl 0718.49013 · doi:10.1007/BF01447329
[12] DOI: 10.1080/17442509408833867 · Zbl 0824.60051 · doi:10.1080/17442509408833867
[13] DOI: 10.1137/S0363012903428664 · Zbl 1101.93086 · doi:10.1137/S0363012903428664
[14] Haussmann U. G., Math. Programming Studies 6 pp 30– (1976)
[15] Ji S., Commun. Inf. Syst. 6 (4) pp 321– (2006)
[16] Kushner H. J., SIAM J. Control Optim. 10 pp 550– (1973)
[17] DOI: 10.1007/BF00353876 · Zbl 0629.60061 · doi:10.1007/BF00353876
[18] DOI: 10.1016/0167-6911(90)90082-6 · Zbl 0692.93064 · doi:10.1016/0167-6911(90)90082-6
[19] DOI: 10.1007/BF01192514 · Zbl 0792.60050 · doi:10.1007/BF01192514
[20] DOI: 10.1137/0328054 · Zbl 0712.93067 · doi:10.1137/0328054
[21] DOI: 10.1007/BF01195978 · Zbl 0769.60054 · doi:10.1007/BF01195978
[22] Peng S., Ser. I 336 pp 773– (2003)
[23] DOI: 10.1137/S0363012996313549 · Zbl 0931.60048 · doi:10.1137/S0363012996313549
[24] Shi J. T., Acta Automatica Sinica 32 (2) pp 161– (2006)
[25] Wu Z., Systems Sci. Math. Sci. 11 (3) pp 249– (1998)
[26] DOI: 10.1017/S0334270000007645 · Zbl 0862.93067 · doi:10.1017/S0334270000007645
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