Study On The Forward-Backward Stochastic Control Systems With Quadratic Generators

The quadratic backward stochastic differential equations(BSDEs)is an important research field of stochastic calculu s.In contrast to the classical BSDEs first studied by Pardoux and Peng in 1990,quadratic BSDEs admit quadratic growth in the argument z,which leads to much difficulty in proving the solvability.Since Kobylanski first studied the quadratic BSDEs with bounded terminal conditions in 2000.many researchers have been made great progress in investigating the well-posedness of quadratic BSDEs with unbounded terminal conditions or whose solutions take values in the multi-dimensional Eucildean spaces.Not only that,quadratic BSDEs also have a wide range of applications in mathematical finance and economics,for instance,the exponential utility maximization problems,risk-sensitive dynamic portfolio management problems.Essentially,these applications arise from a class of stochastic recursive optimal problems in which the controlled quadratic BSDEs are involved.The term "stochastic recursive" originates from the stochastic differential recursive utility studied by Duffie and Epstein in 1992,which can be characterized by the solution of BSDEs.This paper is concerned with the stochastic recursive optimal control problems where the controlled systems are described by forward-backward stochastic differential equations(FBSDEs),while the backward state equations in such controlled FBSDEs are quadratic BSDEs.The research content of this paper consists of the following three parts.In the first part,we study the case where the quadratic BSDE in the controlled FBSDEs possess bounded terminal conditions and generators uniformly bounded in the arguments(x,u).This part corresponds to Chapter 3 and 4 of this paper.We focus on deriving the necessary condition of the optimality-global stochastic maximum principle for this case with general control domains.In order to establish the first-and second-order variational and adjoint equations,we obtain a new estimate for one-dimensional linear BSDEs with unbounded stochastic Lipschitz coefficients involving bounded mean oscillation martingales(BMO martingales)and prove the solvability for a class of multidimensional linear BSDEs with unbounded coefficients but specific structures.Finally,a new global stochastic maximum principle is deduced.As a complement of the above problem,we also investigate the case where the forward state is constrained in a convex set at the torminal time.We use terminal perturbation approach to ovtain a stochastic maximum principle if we regard the terminal state as an admissible control.Is application to a robust portfolio selection problem is also studied.In the second part.we study the case where the quadratic BSDE in the controlled FBSDEs possess unbounded terminal conditions and generators no longer bounded in the arguments(x,u).This part corresponds to Chapter 5 of this paper.Since it is almost impossible for doeision-makers to hold the same risk attitude towards different risk sources in reality,we propose a risk-sensitive criterion of asymmetric risk attitude.Unlike most risk-sensitive literature,the introduced risk-sensitive criterion can only be defined through a quadratic BSDE due to the appearance of asymmetric risk aversion.so it is essentially a quadratic filtration-consistent nonlinear expectation.A mean-variance representation for the introduced risk-sensitive criterion is uncovered by applying a variational approach.We study the linear-quadratic control problem immediately,the optimal feedback control is derived explicitly by the complection of squares technique and Girsanov’s transformation.In addition,a dynamic asset management problem featuring a stochastic return rate and asymmetric risk attitudes is provided as an application.In the third part,we focus on finding approximate solutions to the stochastic recursive optimal control problems where the BSDE in the controlled FBSDEs actually have linear growth in z(of course,this is a special kind of quadratic growth).This part corresponds to Chapters 6 and 7 of this paper.For the first time,we overcome the problem of establishing the modified successive approximation algorithms(MSA)for stochastic recursive optimal control problems with non-convex control domains,and creatively apply the harmonic analysis technique on the space of BMO martingale to the proof of error estimates of MSA algorithms.It breaks through the strong restriction that B.Kerimkulov and his coauthors rely on when proving the error estimates in the case of convex control domains.In the specific case,we obtain a logarithmic convergence rate.When the control domain is convex and compact.a sufficient condition which makes the control returned from the MSA algorithm be a near-optimal control is given for a class of linear controlled FBSDEs.Second.in Chapter 7,we study a classical but important case where the BSDE in the controlled FBSDEs are independent of(y,z).An algorithm based on another MSA is described for finding a set of small measure.in which the control is varied finitely so as to reduce the value of the cost functional.As the control domains are not necessarily convex,the second-order adjoint processes are introduced in each minimization step of the Hamiltonian.Under certain convexity conditions,we prove that the values of the cost functional descend to the global minimum as the number of iterations tends to infinity.In particular,a convergence rate for a class of linear-quadratic systems is available.