Stochastic Linear-Quadratic Optimal Control Theory: Differential Games and Mean-Field Problems by Jingrui Sun & Jiongmin Yong

Author:Jingrui Sun & Jiongmin Yong
Language: eng
Format: epub
ISBN: 9783030483067
Publisher: Springer International Publishing

(b)By a similar method used in proving Proposition 2.1.5 of [48, Chap. 2], one can show that the condition (ii) in Definition 2.6.2 is equivalent to the following:

(ii) for any and ,

Let

Consider the state equation

and the performance functional

where

From (ii) of Remark 2.6.3, we see that is a closed-loop saddle point of Problem (SDG) if and only if is an open-loop saddle point for the problem with the above state equation and performance functional. Applying the idea used in the proof of Theorem 2.5.2, we see that is a closed-loop saddle point of Problem (SDG) if and only if for any , the adapted solution to the FBSDE

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