正倒向随机微分方程理论基础及相关应用  

Theory of Forward Backward Stochastic Differential Equations and Its Applications

作　　者：吴臻[1] 张德涛 WU Zhen;ZHANG Detao(School of Mathematics,Shandong University,Jinan,250100,China;School of Economics,Shandong University,Jinan,250100,China)

机构地区：[1]山东大学数学学院,济南250100 [2]山东大学经济学院,济南250100

出　　处：《应用概率统计》2023年第3期413-435,共23页Chinese Journal of Applied Probability and Statistics

基　　金：国家自然科学基金项目(批准号:11831010、61961160732、12271305);山东省自然科学基金重大基础研究项目(批准号:ZR2019ZD42、ZR2020ZD24)资助.

摘　　要：本文从随机微分方程和倒向随机微分方程基本理论和应用背景谈起,结合随机最优控制理论和金融市场中的期权定价理论导出完全耦合的正倒向随机微分方程的形式.进而从该类方程的可解性这一角度出发,对已有的理论方法进行分析和探讨,引入一种非马尔科夫框架下保证解的存在唯一性的“统一框架”方法,给出比较定理、解的高维估计等重要性质,并联系相关偏微分方程系统给出其概率解释.对实际中应用广泛的线性正倒向随机微分方程引入了一种线性变换的方法作为“统一框架”方法的重要补充和完善,使得正倒向随机微分方程的应用更加广泛.In this paper,based on the basic theory and application background of stochastic differential equation and backward stochastic differential equation,and combined with stochastic optimal control theory and option price theory in financial market,we will derive the general form of fully coupled forward backward stochastic differential equations(FBSDEs in short).From the point of view of the solvability of this kind of equations,the existing methodology in the literature are analyzed and discussed,a“unified framework”approach is introduced to guarantee the existence and uniqueness of solutions for non-Markovian FBSDEs,and several further properties of FBSDEs are obtained.A linear transformation method in virtue of the non-degeneracy of transformation matrix is introduced for cases that the linear FBSDEs,as an important supplement and improvement of the“unified approach”method,which makes the application of FBSDEs more extensive.

关 键 词：倒向随机微分方程 正倒向随机微分方程 随机控制 统一框架方法 偏微分方程 

分 类 号：O231.3[理学—运筹学与控制论] O211.6[理学—数学] F224.7[经济管理—国民经济]