平均场随机控制及微分对策

For the future works the applicant will continue the study on the theory of general mean field (forward) backward stochastic differential equations (SDEs) (i.e., the coefficients depend not only on the solution but also the law induced by the solution), and its applications in stochastic control, stochastic differential game, partial differential equations and mathematical finance. A particular attention among these future works will be paid to challenging problems in the interface between classical stochastic control and game theory on one hand and, on the other hand, the modelization of mean field interaction (a very recent research topic motivated by a large number of involved agents or players). Related subjects include: 1. The study of viscosity solutions of non-linear partial differential equations (PDEs) over spaces of probability measures-a completely new type of PDEs (including HJB equations and HJB-Isaacs equations)-with the help of general (forward) backward SDEs of mean field type; 2. The existence of the solution and the comparison theorems of general (forward) backward SDEs of mean field type under non-Lipschitz conditions; 3. Representation of limit values of Abel mean and Cesaro mean for nonexpansive stochastic differential games and also nonexpansive stochastic differential control in the mean field case. . The applicant has achieved different research results of high international level in the fields of stochastic differential games, forward-backward stochastic differential equations, stochastic control, and mathematical finance. 33 SCI papers were published including 20 SCI papers during the recent five years. Our papers have been cited 272 times by SCI papers of other authors between 2013 and 2017. Our papers until now have been commented, generalized and cited by a lot of international famous experts, among them, the famous French academicians Professors P.L. Lions (who is also the winner of Fields Prize) and A. Bensoussan (who is also one of the founders of the modern stochastic control theory and former chief of the French Space Research Center); the outstanding American mathematicians Professors W.H. Fleming and N.V. Krylov. The applicant has got the Excellent Youth Fund of National Natural Science, the fund from the program for New Century Excellent Talents in University, the Distinguished Youth Fund of Natural Science of Shandong Province, as well as a Newton Advanced Fellowship.

本项目旨在以随机分析中的(正)倒向随机微分方程理论为基础，深入研究一般情形下的平均场(正)倒向随机微分方程(即，此时方程的系数依赖于解以及解引出的分布)，及其在随机控制、随机微分对策、偏微分方程、金融等方面的应用，具体来说: 1.利用一般情形下的平均场(正)倒向随机微分方程研究新型的定义在概率集合上的非局部偏微分方程(包括相应的Hamilton-Jacobi-Bellman(-Isaacs)方程)的粘性解的存在唯一性；2.研究一般情形下的平均场(正)倒向随机微分方程在各种非Lipschitz条件下的解的存在性以及比较定理；3.研究(非扩张条件下)随机微分对策问题以及平均场情形下的随机控制问题中的Abel均值和Cesaro均值的极限的表示定理等。得到一批在相关领域居于国际前沿的成果。

本项目主要研究内容为(平均场)正倒向随机微分方程及其在随机控制、随机微分对策、偏微分方程和金融数学等领域中的应用。既是重要的理论研究课题，又有着很强的应用前景。.项目执行期间取得的部分重要结果如下：1)研究了连续系数下一般情形下的平均场倒向重随机微分方程。通过对测度引入一个新的加权度量，首次得到了对方程的倒向积分中关于Z的分布的Lipschitz常数没有任何限制的解的存在唯一性等结论。进而得到了连续系数下该方程解的存在性和比较定理等。2)研究了一般情形下的高维平均场倒向重随机微分方程的比较定理及其在连续系数下的解的存在性等。3)首次研究了一般情形下依赖于条件分布的平均场随机微分方程，用全新的方法证明了当系数有界可测，关于状态和测度满足Lipschitz条件时，方程弱解存在且依分布唯一。4)首次研究了非扩张条件下正倒向随机微分方程的随机微分对策问题的值函数的极限以及表示定理。不同于遍历控制理论中，该极限可以是函数，依赖于初始状态。并把该极限刻画为相应的极限偏微分方程的最大粘性下解等。5)首次研究了随机控制问题中的Abel均值和Cesàro均值的极限值的表示定理，并对其极限值做了比较。分别借助于占位测度和松弛控制给出了它们的聚点的两种表示，并证明了若其聚点存在必相等。进一步，证明了在非扩张条件下Abel均值和Cesàro均值的极限存在并相等。6)首次研究了耦合的高维的HJB-Isaacs方程粘性解的概率表示等。.项目执行期间项目负责人李娟已发表文章10篇，发表的期刊包括Siam. J. Control. Optim.；Journal of Differential Equations；Appl. Math. Optim.；Applied Math. Computation；ESAIM: COCV；Stochastic Proc. Appli.；J. Math. Anal. Appl.；SIAM Journal on Financial Math.。1篇文章被Annals of Applied Probability接收。已投稿3篇。到目前为止已取得很多重要的国际领先的成果，而这些成果为进一步研究平均场随机动力系统及其应用奠定了基础。项目负责人积极参加国际国内会议及访问交流报告相关成果，同时也邀请了国际国内专家访问交流作报告。这些学术活动促进了本项目的顺利实施。