Generalized Mean-field Backward Stochastic Differential Equations And Variational Inequalities And Related Optimal Control Problems

The generalized backward stochastic differential equations,compared with the general backward stochastic differential equations,have an extra integral with respect to the increasing process,which also can be applied to study optimal control problems.We can use them to find the optimal solution from all possible control schemes.The generalized backward stochastic differential equation was first studied by Pardoux and Zhang[1]in 1998,they also studied the related partial differential equations with nonlinear Neumann boundary conditions.In 2015,Li and Tang[2]studied the optimal control problem of stochastic dynamic reflected in a domain with the recursive cost functionals.In 2017,Pardoux and Rǎscanu[3]studied the continuity of the solution for a generalized parabolic equation with nonlinear Neunmann boundary condition.In 2020,Feng[4]studied the generalized mean-field backward stochastic differential equations and related partial differential equations with nonlinear Neumann boundary conditions.Based on above works,we mainly study generalized backward stochastic differential equation and generalized backward stochastic variational inequality in the mean-field case to investigate the properties of their solutions,related partial differential equation problems,and their application in optimal control problems.Our paper is divided into the following three parts.In the first part(Chapter 3),we study the existence and the uniqueness of the solutions of generalized mean-field backward stochastic differential equation and its comparison theorem,that is,the coefficient of the equation depends not only on the state of the solution,but also on the distribution of the solution nonlinearly.We consider the equation as follows:where K is a real value continuous increasing process that satisfies certain integrability conditions.With Lipschitz condition and monotonicity condition we prove the existence and the uniqueness of the solution of the equation(0-1)by using the fixed point theorem,and prove the corresponding comparison theorem with the help of the It(?)-Tanaka formula.On the other hand,we give two examples on the comparison theorem to explain the necessity that K should be the same and the coefficient f does not depend on the law of Z.In the second part(Chapter 4),for any(t,x)∈[0,T]× D and(x0,u)∈ D × Ut,T,we consider the following mean-field decoupled FBSDE:where X0,x0;u,Xt,x;u is the solution of the following reflected SDE,respectively:where Ut,T denotes the admissible control set on[t,T],D is an open connected bounded convex subset of Rd,W(t,x):=(?)Ytt,x;u is the value function of the control problem.We consider the following HJB equation with nonlinear Neumann boundary conditions:where at x∈(?)D,(?)X0,x0;u is the solution of SDE(0-3)with initial data(0,x0)and given control u ∈ U0,T,and Hamilton function H is defined as:where(t,x,y,p,A)∈[0,T]× Rd × R ×Rd × Sd,Sd is the set of all d × d dimensional symmetric matrix.With Lipschitz condition and monotonicity condition we use the Picard iteration to prove the existence and the uniqueness of GBSDE(0-2)and its comparison theorems,and prove that the value function W is the unique viscosity solution of HJB equation(0-4)with the help of dynamic programming principle.In the third part(Chapter 5),we consider the following generalized mean-field backward stochastic variational inequality:where K is a real value continuous increasing process that satisfies certain integrability conditions.We study the existence and the uniqueness of the solution to the inequality(0-5)and the corresponding comparison theorems.We define random field:Furthermore,we consider the following PVI system:where operator lt is defined as:and φ,ψ:Rm(?)(-∞,+∞]in(0-6)are proper convex lower semicontinuous functions.The multivalued subdifferential operator (?)φ is defined as:(?)ψ is defined similarly.Under Lipschitz condition and monotonicity condition,we prove that u is continuous with respect to(t,x).And we use the Ishii Lemma to prove that u is the unique viscosity solution of(0-6).