正倒向随机微分方程及相关的优化问题

In this project, the core of our research is the coupled forward-backward stochastic differential equations (FBSDEs for short). We shall study the following problems systematically and deeply. (1) We shall improve the Lp (p is bigger than or equal to 2) theory for the coupled FBSDEs. Then, in the sense of classical solutions and weak solutions, we shall investigate the probabilistic interpretation for a kind of second order quasilinear partial differential equations. (2) By delicately using the Lp theory of the coupled FBSDEs, we study the stochastic optimal control problems and the stochastic differential game problems, in which the recursive indices are coupled with the states. We shall generalize the dynamic programing principle and the theory of viscosity solution for the related Hamilton-Jacobi-Bellman equation, and establish a general maximum principle. (3) By clearly analyzing the coupled structure of FBSDEs, we propose a kind of linear-quadratic generalized stochastic Stackelberg games. Moreover, for the Stackelberg equilibrium, we shall study the existence, the uniqueness, the open-loop form and the close-loop form. (4) Finally, the theoretical results will be applied to several problems arising from the field of mathematical finance, such as the investment optimization problems and the asset pricing problems.

本课题以耦合的正倒向随机微分方程为核心，系统深入地研究下面问题：（1）改进耦合的正倒向随机微分方程的Lp（p大于等于2）理论，在经典解和弱解意义下，全面深入地研究一类二阶拟线性偏微分方程的概率解释；（2）精细地使用耦合的正倒向随机微分方程的Lp理论，研究递归型的指标和状态耦合的随机最优控制和随机微分博弈问题，包括推广动态规划原理和相关的Hamilton-Jacobi-Bellman方程粘性解理论，建立一般最大值原理；（3）清晰地分析正倒向随机微分方程的耦合结构，提出线性二次广义随机Stackelberg博弈，并研究Stackelberg均衡的存在性、唯一性、开环和闭环表达；（4）将理论结果应用于若干金融数学领域的投资优化和资产定价问题。

本课题研究了以耦合的正倒向随机微分方程为核心的随机最优控制和随机微分博弈理论及相关问题。主要包括：改进了耦合的正倒向随机微分方程的Lp（p≥2）理论，得到小区间上的Lp结果；发展了正倒向随机微分方程的自生长、自相似的“控制—单调结构”，拓宽了广义 Stackelberg 随机微分博弈的研究领域；研究了有关均值场、无穷时区、时间延迟、随机跳跃等数学模型下的线性二次最优控制和零和微分博弈问题；研究了非线性期望框架下的倒向随机微分方程理论；发展了随机系统的能控性的研究。除此之外，我们解决了若干金融数学领域的投资优化问题，推动随机控制与博弈、随机分析、金融数学等学科领域的发展。