Maximum principle for forward-backward doubly stochastic control systems and applications

Liangquan Zhang; Yufeng Shi

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 4, page 1174-1197
  • ISSN: 1292-8119

Abstract

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The maximum principle for optimal control problems of fully coupled forward-backward doubly stochastic differential equations (FBDSDEs in short) in the global form is obtained, under the assumptions that the diffusion coefficients do not contain the control variable, but the control domain need not to be convex. We apply our stochastic maximum principle (SMP in short) to investigate the optimal control problems of a class of stochastic partial differential equations (SPDEs in short). And as an example of the SMP, we solve a kind of forward-backward doubly stochastic linear quadratic optimal control problems as well. In the last section, we use the solution of FBDSDEs to get the explicit form of the optimal control for linear quadratic stochastic optimal control problem and open-loop Nash equilibrium point for nonzero sum stochastic differential games problem.

How to cite

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Zhang, Liangquan, and Shi, Yufeng. "Maximum principle for forward-backward doubly stochastic control systems and applications." ESAIM: Control, Optimisation and Calculus of Variations 17.4 (2011): 1174-1197. <http://eudml.org/doc/272796>.

@article{Zhang2011,
abstract = {The maximum principle for optimal control problems of fully coupled forward-backward doubly stochastic differential equations (FBDSDEs in short) in the global form is obtained, under the assumptions that the diffusion coefficients do not contain the control variable, but the control domain need not to be convex. We apply our stochastic maximum principle (SMP in short) to investigate the optimal control problems of a class of stochastic partial differential equations (SPDEs in short). And as an example of the SMP, we solve a kind of forward-backward doubly stochastic linear quadratic optimal control problems as well. In the last section, we use the solution of FBDSDEs to get the explicit form of the optimal control for linear quadratic stochastic optimal control problem and open-loop Nash equilibrium point for nonzero sum stochastic differential games problem.},
author = {Zhang, Liangquan, Shi, Yufeng},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {maximum principle; stochastic optimal control; forward-backward doubly stochastic differential equations; spike variations; variational equations; stochastic partial differential equations; nonzero sum stochastic differential game; optimal control problems; fully coupled forward-backward doubly stochastic differential equations (FBDSDEs); stochastic maximum principle (SMP); stochastic partial differential equations (SPDEs)},
language = {eng},
number = {4},
pages = {1174-1197},
publisher = {EDP-Sciences},
title = {Maximum principle for forward-backward doubly stochastic control systems and applications},
url = {http://eudml.org/doc/272796},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Zhang, Liangquan
AU - Shi, Yufeng
TI - Maximum principle for forward-backward doubly stochastic control systems and applications
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2011
PB - EDP-Sciences
VL - 17
IS - 4
SP - 1174
EP - 1197
AB - The maximum principle for optimal control problems of fully coupled forward-backward doubly stochastic differential equations (FBDSDEs in short) in the global form is obtained, under the assumptions that the diffusion coefficients do not contain the control variable, but the control domain need not to be convex. We apply our stochastic maximum principle (SMP in short) to investigate the optimal control problems of a class of stochastic partial differential equations (SPDEs in short). And as an example of the SMP, we solve a kind of forward-backward doubly stochastic linear quadratic optimal control problems as well. In the last section, we use the solution of FBDSDEs to get the explicit form of the optimal control for linear quadratic stochastic optimal control problem and open-loop Nash equilibrium point for nonzero sum stochastic differential games problem.
LA - eng
KW - maximum principle; stochastic optimal control; forward-backward doubly stochastic differential equations; spike variations; variational equations; stochastic partial differential equations; nonzero sum stochastic differential game; optimal control problems; fully coupled forward-backward doubly stochastic differential equations (FBDSDEs); stochastic maximum principle (SMP); stochastic partial differential equations (SPDEs)
UR - http://eudml.org/doc/272796
ER -

References

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