scholarly journals Fully Coupled Mean-Field Forward-Backward Stochastic Differential Equations and Stochastic Maximum Principle

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Hui Min ◽  
Ying Peng ◽  
Yongli Qin
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Optimal Control Problems ◽  
Mean Field ◽  
Backward Stochastic Differential Equations ◽  
Maximum Principles ◽  
Control Problems ◽  
Linear Quadratic Optimal Control ◽  
Fully Coupled

We discuss a new type of fully coupled forward-backward stochastic differential equations (FBSDEs) whose coefficients depend on the states of the solution processes as well as their expected values, and we call them fully coupled mean-field forward-backward stochastic differential equations (mean-field FBSDEs). We first prove the existence and the uniqueness theorem of such mean-field FBSDEs under some certain monotonicity conditions and show the continuity property of the solutions with respect to the parameters. Then we discuss the stochastic optimal control problems of mean-field FBSDEs. The stochastic maximum principles are derived and the related mean-field linear quadratic optimal control problems are also discussed.

scholarly journals Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations

2011 ◽  
Vol 43 (02) ◽  
pp. 572-596 ◽  
Author(s):  
Bernt Øksendal ◽  
Agnès Sulem ◽  
Tusheng Zhang
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Optimal Control Problems ◽  
Backward Stochastic Differential Equation ◽  
Maximum Principles ◽  
Optimal Consumption ◽  
Control Problems ◽  
Stochastic Delay ◽  
Stochastic Delay Equations

We study optimal control problems for (time-)delayed stochastic differential equations with jumps. We establish sufficient and necessary stochastic maximum principles for an optimal control of such systems. The associated adjoint processes are shown to satisfy a (time-)advanced backward stochastic differential equation (ABSDE). Several results on existence and uniqueness of such ABSDEs are shown. The results are illustrated by an application to optimal consumption from a cash flow with delay.

scholarly journals Stochastic Optimization Theory of Backward Stochastic Differential Equations Driven by G-Brownian Motion

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Zhonghao Zheng ◽  
Xiuchun Bi ◽  
Shuguang Zhang
Keyword(s):  
Optimal Control ◽  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Optimal Control ◽  
Optimal Control Problems ◽  
Backward Stochastic Differential Equations ◽  
Dynamic Programming Principle ◽  
Control Problems ◽  
Order Partial Differential Equation

We consider the stochastic optimal control problems under G-expectation. Based on the theory of backward stochastic differential equations driven by G-Brownian motion, which was introduced in Hu et al. (2012), we can investigate the more general stochastic optimal control problems under G-expectation than that were constructed in Zhang (2011). Then we obtain a generalized dynamic programming principle, and the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.

scholarly journals Optimal control of stochastic delay equations and time-advanced backward stochastic differential equations

2011 ◽  
Vol 43 (2) ◽  
pp. 572-596 ◽  
Author(s):  
Bernt Øksendal ◽  
Agnès Sulem ◽  
Tusheng Zhang
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Optimal Control Problems ◽  
Backward Stochastic Differential Equation ◽  
Maximum Principles ◽  
Optimal Consumption ◽  
Control Problems ◽  
Stochastic Delay ◽  
Stochastic Delay Equations

We study optimal control problems for (time-)delayed stochastic differential equations with jumps. We establish sufficient and necessary stochastic maximum principles for an optimal control of such systems. The associated adjoint processes are shown to satisfy a (time-)advanced backward stochastic differential equation (ABSDE). Several results on existence and uniqueness of such ABSDEs are shown. The results are illustrated by an application to optimal consumption from a cash flow with delay.

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