Infinite Horizon Backward Stochastic Differential Equations Under Nonlinear Expectations And Related Topics

In the classical probability framework,the theory of nonlinear backward stochastic differential equations(BSDEs)on finite horizon,established by Pardoux and Peng in 1990,provides a powerful tool for several fields such as partial differential equations(PDEs),stochastic control and mathematical finance.It has been widely concerned by many researchers about how to generalize the classical finite horizon BSDEs under Lipschitz condition and apply them to various problems.However,it is usually difficult to find an ideal situation with specific probability when we treat the problems in the real world because of the uncertainty of the probability itself.In economics,this type of uncertainty is known as Knightian uncertainty.Motivated by this,Peng established the axiomatic definition of consistent nonlinear expectations and constructed a typical consistent sublinear expectation called G-expectation as well as G-Brownian motion and the corresponding G-It? stochastic calculus.An important problem is to establish the theory of BSDEs under the nonlinear expectation framework which would provide efficient methods to problems under model uncertainty like classical BSDEs in probability space.In this thesis,we study the infinite horizon BSDEs and related problems under consistent nonlinear expectations.We generalize the existing results to general nonlinear expectation framework as well as the case when generators are quadratic,and also give an application in mathematical finance.In addition,we also investigate the property of G-normal distribution which is an important concept in the G-expectation framework.The main contribution of this thesis is that we develop the theory of infinite horizon BSDEs under consistent nonlinear expectations and establish the wellposedness of this type of BSDEs under different settings and assumptions.In particular,we propose some new methods to deal with the difficulties arising from different conditions.Moreover,we also give the representation for a new type of robust forward performance processes via infinite horizon G-BSDEs.There are seven chapters herein.Chapters 1 and 2 are the introduction and preliminaries,respectively,and Chapter 3 gives the property of G-normal distribution as an extension of Chapter 2.In Chapters 4-6 we investigate infinite horizon BSDEs under consistent nonlinear expectations in different settings,and the final chapter summarizes.The thesis is organized as follows:In Chapter 1,we introduce the history of the development of consistent nonlincar expectations and BSDEs,as well as the innovations and the structure of this thesis,while we state some basic definitions and related results in consistent nonlinear expectation framework(especially G-expectation framework)in Chapter Chapter 3 discusses the explicit solutions to G-heat equations and the property of G-normal distribution as an extension of Chapter 2.More precisely,we study a.class of explicit positive solutions to G-heat equations by solving second order nonlinear ordinary differential equations and give the sharp order of G-capacity of G-normal distribution in a small ball based on the positive explicit solutions.Chapter 4 considers the framework of general consistent nonlinear expectations which are dominated by consistent sublinear expectations.We obtain the explicit solutions to linear BSDEs,priori estimates and linearization method by some new methods and then establish the wellposedness of infinite horizon BSDEs under this consistent nonlinear expectation framework.Besides,we show the existence of the solutions of Markovian ergodic BSDEs under nonlinear (?)-expectations and state some applications to PDEs.Chapter 5 establishes the existence and uniqueness result.for infinite horizon quadratic G-BSDEs.With the help of G-BMO martingale generators and corresponding G-Girsanov transformation,we give the explicit solutions to linear G-BSDEs with unbounded coefficients and linearization method for quadratic generators and then construct the unique solutions to infinite horizon quadratic G-BSDEs.In Chapter 6,we show the wellposedness of infinite horizon Markovian quadratic G-BSDEs as well as Markovian quadratic ergodic G-BSDEs via truncation method.As an application,we define a new type of robust forward performance processes called G-forward performance processes,and derive the representations for homothetic(exponential,power,logarithmic)G-forward performance processes via these two types of BSDEs.In the last chapter,we briefly give the conclusion of this thesis and discuss some prospects for our future work.