Stochastic maximum principle for delayed backward doubly stochastic control systems

Volume 10, Issue 1, pp 215--226 http://dx.doi.org/10.22436/jnsa.010.01.21

Publication Date: January 27, 2017 Submission Date: June 10, 2016

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Authors

Jie Xu - School of Mathematics, Jilin University, Changchun, 130012, China. - College of Sciences, Jilin Institute of Chemical Technology, Jilin, 132022, China. Yuecai Han - School of Mathematics, Jilin University, Changchun, 130012, China.

Abstract

In this paper, we investigate a class of doubly stochastic optimal control problems that the state trajectory is described by backward doubly stochastic differential equations with time delay. By means of martingale representation theorem and contraction mapping principle, the existence and uniqueness of solution for the delayed backward doubly stochastic differential equation can be guaranteed. When the control domain is convex, we deduce a stochastic maximum principle as a necessary condition of the optimal control by using classical variational technique. At the same time, under certain assumptions, a sufficient condition of optimality is obtained by using the duality method. In the last section, we give the explicit form of the optimal control for delayed doubly stochastic linear quadratic optimal control problem by our stochastic maximal principle.

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Keywords

• Stochastic maximum principle
• doubly stochastic differential equation
• time delay
• optimal control.

MSC

•  60H10
•  93E20

References

• [1] L. Chen, J.-H. Huang, Stochastic maximum principle for controlled backward delayed system via advanced stochastic differential equation, J. Optim. Theory Appl., 167 (2015), 1112–1135.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] L. Chen, Z. Wu, Maximum principle for the stochastic optimal control problem with delay and application, Automatica J. IFAC, 46 (2010), 1074–1080.
  • View Article
  • MathSciNet
  • Google Scholar


• [3] M. Hafayed, Singular mean-field optimal control for forward-backward stochastic systems and applications to finance, Int. J. Dyn. Control, 2 (2014), 542–554.
  • View Article
  • Google Scholar


• [4] M. Hafayed, M. Ghebouli, S. Boukaf, Partial information optimal control of mean-field forwardbackward stochastic system driven by Teugels martingales with applications, Neurocomputing, 200 (2016), 11–21.
  • View Article
  • Google Scholar


• [5] M. Hafayed, M. Tabet, S. Boukaf, Mean-field maximum principle for optimal control of forward-backward stochastic systems with jumps and its application to mean-variance portfolio problem, Commun. Math. Stat., 3 (2015), 163–186.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] Y.-C. Han, S.-G. Peng, Z. Wu, Maximum principle for backward doubly stochastic control systems with applications, SIAM J. Control Optim., 48 (2010), 4224–4241
  • View Article
  • MathSciNet
  • Google Scholar


• [7] J.-H. Huang, X. Li, J.-T. Shi, Forward-backward linear quadratic stochastic optimal control problem with delay, Systems Control Lett., 61 (2012), 623–630.
  • View Article
  • MathSciNet
  • Google Scholar


• [8] Q. Lin, A generalized existence theorem of backward doubly stochastic differential equations, Acta Math. Sin. (Engl. Ser.), 26 (2010), 1525–1534.
  • View Article
  • MathSciNet
  • Google Scholar


• [9] A. Matoussi, M. Scheutzow, Stochastic PDEs driven by nonlinear noise and backward doubly SDEs, J. Theoret. Probab., 15 (2002), 1–39.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] D. Nualart, E. Pardoux , Stochastic calculus with anticipating integrands, Probab. Theory Related Fields, 78 (1988), 535–581.
  • View Article
  • MathSciNet
  • Google Scholar


• [11] É. Pardoux, S.-G. Peng, Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Related Fields, 98 (1994), 209–227.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] S.-G. Peng, Y.-F. Shi, A type of time-symmetric forward-backward stochastic differential equations, C. R. Math. Acad. Sci. Paris, 336 (2003), 773–778.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] S.-G. Peng, Z. Yang, Anticipated backward stochastic differential equations, Ann. Probab., 37 (2009), 877–902.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] D. Revuz, M. Yor, Continuous martingales and Brownian motion , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin (1991)
  • Google Scholar


• [15] Y.-F. Shi, Y.-L. Gu, K. Liu, Comparison theorems of backward doubly stochastic differential equations and applications, Stoch. Anal. Appl., 23 (2005), 97–110.
  • View Article
  • MathSciNet
  • Google Scholar


• [16] S. Wu, G.-C. Wang, Optimal control problem of backward stochastic differential delay equation under partial information, Systems Control Lett., 82 (2015), 71–78.
  • View Article
  • MathSciNet
  • Google Scholar


• [17] X.-M. Xu, Anticipated backward doubly stochastic differential equations, Appl. Math. Comput., 220 (2013), 53–62.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] J.-M. Yong, Optimality variational principle for controlled forward-backward stochastic differential equations with mixed initial-terminal conditions, SIAM J. Control Optim., 48 (2010), 4119–4156.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] F. Zhang, Anticipated backward doubly stochastic differential equations, (Chinese) J. Sci. Sin. Math., 43 (2013), 1223– 1236.
  • View Article
  • Google Scholar