The Adaptive Solutions Of FBSDE And Their Application To Stochastic Differential Utility

The study of forward-backward stochastic differential equations stems from the problems on stochastic control and finance. Conversely, the theoretical results of it are widely applied to the fields of stochastic control and finance as well as partial differential equations. Based on the research of forward and backward stochastic differential equations respectively, the study of FBSDE progressed rapidly but insufficiently, as most of the results of FSDE and BSDE cannot be applied to FBSDE directly.At present, the study of FBSDE can be separated into two major trends: l.the backward part is driven by Brown Motion (called Ito type); 2.the backward part is driven by conditional expectation with natural filtration under ordinary conditions. Because of the difference in filtration and integrability of the coefficients, these two types are not closely inter-related.So far, there is no systematic result on the adaptive solutions of FBSDE of the second type. Consequently, the related comparison theorem, properties on SDU and stochastic control are not well explored. Meanwhile, the lack of study of FBSDE under local Lipschitz condition results in much inconvenience in its application to finance.This paper mostly focuses on the existence and uniqueness ofthe adaptive solutions of FBSDE(type 2) and theirapplication to SDU. The main results are: In chapter 2, underthe system driven by increasing processes and using the Itoformula, optional projection theorem and contraction maping,we can get the comparison theorem of certain FBSDE and thecorresponding properties on SDU, such as monotonicity,concavity and risk aversion; in chapter 3, when the terminalis stopping time, we can also get the existence anduniqueness of the adaptive solutions of these FBSDE, andtheir application to the infinite horizon SDU; in chapter 4,under some special monotonicity condition, the existenceand uniqueness of the adaptive solutions can easily beobtained with a probabilistic method to treat a large kind ofFBSDE with an arbitrarily prescribed time duration; inchapter 5, the solvability of some FBSDE and the stability of the solutions under local Lipschitz condition is discussed; and finally in chapter 6, a maximum principle for optimal control problem of the FBSDE system in chapter 4 is achieved. Throughout the chapters, contraction maping, ltd formula and optional projection are widely applied to the proof of existence and uniqueness of the adaptive solutions of FBSDE.