【作者】 左姗姗；

【导师】 李娟；

【作者基本信息】 山东大学 ， 运筹学与控制论， 2013， 硕士

【摘要】 自从Buckdahn, Djehiche, Li和Peng [1]首次将平均场引入到倒向随机微分方程(BSDEs)之后,平均场倒向随机微分方程(mean-field backward stochastic differential equations (Mean-field BSDEs))便受到了广大学者的关注。基于此,本文将研究平均场正倒向随机系统的微分对策问题：其中λv1,v2(t)=(xv1,v2(t),yv1,v2(t),zv1,v2(t))。在适当的假设下,本文引入相应的代价泛函,研究该平均场正倒向随机系统的微分对策问题：零和以及非零和微分对策的最大值原理。给出了最大值原理的必要条件与充分条件,并且通过给出一非零和微分对策的例子来进一步阐述本文主要定理的应用。由于一般的随机控制问题可以视为只有一个参赛者的零和微分对策问题,并且在现实生活中,观测者多数情况下只能得到部分信息,所以本文首先研究了部分信息下的平均场正倒向随机微分方程(mean-field forward-backward stochastic differential equations (Mean-field FBSDEs))的最优控制问题,并且利用经典的凸变分技术得到了最优控制的必要条件,其中,随机系统描述如下：更多还原

【Abstract】 In2009, Buckdahn, Djehiche, Li and Peng [1] firstly introduced the mean-field theory to backward stochastic differential equations (BSDEs), and obtain-ed a new type of backward stochastic differential equations--Mean-field backw-ard stochastic differential equations (Mean-field BSDEs). We will study the di-fferential games of mean-field forward-backward stochastic systems:where λv1,v2(t)=(xv1,v2(t),yv1,v2(t),zv1,v2(t)).Under appropriate assumptions, this paper mainly works on the maximum principle for both zero-sum and nonzero-sum games. We give a necessary con--dition and a sufficient condition in the form of maximum principle for the games. In the end, we give an example of a nonzero-sum game of mean-field FBSDE to explain our main results.Since the general stochastic control problems could be regarded as the zero-sum differential games with only one player, and in reality, the observers can only observe the partial information which is a sub-filtration in probability lang--uage, we will firstly focus on the optimization problems of mean-field FBSDEs with partial information. With the help of the classical convex variational techn--ique, we establish a necessary maximum principle for the optimization proble--ms, where the stochastic system is described as follows:更多还原