Decoupling Method For General Linear FBSDEs And Its Applications

The decoupling solution of forward and backward stochastic differential equations(FBSDEs)is the basis of optimal control theory.The solvability of the optimal control is divided into two parts:firstly,the maximum principle obtained by variational method,which is the FBSDEs under the equilibrium condition;Secondly,solving the FBSDEs to design the optimal controllers.Therefore,it is of great significance to solve the FBSDEs.The research began in the 1970s and has made important progress over the past half century.However,it is noted that the existing results are mainly for the special FBSDEs,and the decoupling solution is still faced with obstacles in the general case with time delay.The main contribution of this paper is to propose a decoupling method for general linear FBSDEs with time delay.By using the method of discretization,backward iterative induction and limit approximation technique,the inhomogeneous relationship between forward and backward stochastic processes is established,the solvable condition and explicit solution of the equations are also given.Another contribution of this paper is to apply the decoupling method to solve stochastic optimal control problems and rational expectations model.For stochastic optimal control problems with transmission delay and state delay,the solvability of the problem is equal to that of the FBSDEs with time delay by using the maximum principle.Based on the proposed decoupling method,the solvability condition,explicit optimal controller and optimal cost function for the control problem are given.Furthermore,the optimal control with asymmetric information and rational expectations model are solved by decoupling method.The main contents and results are listed as follows.1.A decoupling method for FBSDEs with input delay is proposed.For the FBSDEs with input delay,the forward backward stochastic difference equations are obtained based on the method of discretization and backward iterative induction.By introducing a kind of Riccati equation,the inhomogeneous relation between the forward and backward stochastic difference equations is established,the solvable condition and explicit solution are obtained.Then the solvability condition and explicit solution of the original differential equation are given by limit approximation technique.2.A decoupling method for FBSDEs with state delay is proposed.For the FBSDEs with state delay,the forward backward stochastic difference equations are obtained based on the method of discretization and backward iterative induction.By introducing a kind of Riccati equation with state delay,the inhomogeneous relation between the forward and backward stochastic difference equations is established,the solvable condition and explicit solution are obtained.Then the solvability condition and explicit solution of the original differential equation are given by limit approximation technique.3.The optimal control problem of transmission delay or state delay are solved based on the decoupling method.The problem is transformed into a time delay FBSDEs,and the explicit solution of the corresponding FBSDEs with time delay is given by combining the decoupling method.Based on the transmission delay and state delay Riccati equations,the solvable conditions and explicit optimal controller of the optimal control problem are obtained.4.The optimal control problem with asymmetric information is solved based on decoupling method.Firstly,combined with the decoupling method,the solvable condition and explicit solution for the FBSDEs with asymmetric information are given.Secondly,the general optimal control problem with asymmetric information is considered,and the problem is transformed into the solvability of FBSDEs with asymmetric information through the maximum principle.Combined with the new information structure by equivalent transformation and the decoupling method.Based on a kind of Riccati equations with asymmetric information,the solvable condition,explicit optimal controllers and optimal cost function of the optimal control problem are given.5.The rational expectations model with long expected time and state delay are solved based on the decoupling method.In the case of long forward looking time,the solvability condition and explicit solution of the problem are given based on the Riccati equation with rational expectations by combining the maximum principle and the decoupling method.For the case of state delay,by transforming into a kind of FBSDEs in finite horizon and infinite horizon with equivalent transformation.Then combining with decoupling method,the solvable condition and explicit solution of the problem are given based on a class of Riccati equations with state delay.