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 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 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.

On the partial controllability of SDEs and the exact controllability of FBSDES

2020 ◽  
Vol 26 ◽  
pp. 68 ◽  
Author(s):  
Yanqing Wang ◽  
Zhiyong Yu
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Optimal Control Problems ◽  
Exact Controllability ◽  
Adjoint Equations ◽  
Control Problems ◽  
Output Controllability ◽  
Time Invariant ◽  
Partial Controllability

A notion of partial controllability (also can be called directional controllability or output controllability) is proposed for linear controlled (forward) stochastic differential equations (SDEs), which characterizes the ability of the state to reach some given random hyperplane. It generalizes the classical notion of exact controllability. For time-invariant system, checkable rank conditions ensuring SDEs’ partial controllability are provided. With some special setting, the partial controllability for SDEs is proved to be equivalent to the exact controllability for linear controlled forward-backward stochastic differential equations (FBSDEs). Moreover, we obtain some equivalent conclusions to partial controllability for SDEs or exact controllability for FBSDEs, including the validity of observability inequalities for the adjoint equations, the solvability of some optimal control problems, the solvability of norm optimal control problems, and the non-singularity of a random version of Gramian matrix.

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