Numerical Algorithm For Optimal Control Problem Of A Class Of Forward-Backward Stochastic Differential Equation

In the mixed optimal investment strategies of financial enterprises,the classical stochastic optimal control systems can not well describe the decision-making problems.The optimal control problem controlled by forward-backward stochastic differential equations(FBSDEs)can better describe the optimal investment strategy problems.But in practical calculation,the structures of the forward-backward stochastic optimal control problems are complex,so it is very difficult to obtain its explicit solution directly by analytical method.Therefore,it is very important to study its numerical algorithm.Based on the FBSDEs and its optimal control theories,a numerical algorithm for solving a class of FBSDEs optimal control problems is discussed in this paper.Firstly,based on the stochastic maximum principle,we obtain the optimal systems of the FBSDESs optimal control problem.Secondly,based on the optimal systems,combined with the explicit differential multistep method of FBSDEs,a gradient projection optimization algorithm for solving the FBSDEs optimal control problem is constructed.Then,based on the convergence error of the FBSDEs’algorithm,the convergence estimation of the whole numerical algorithm is derived.Finally,the theoretical results are verified by numerical experiments.In addition,when analyzing the error of the FBSDEs optimal control,it is necessary to use the convergence conclusion of FBSDEs’ numerical method.Although a large number of numerical examples verify the stability and convergence of FBSDEs high-precision method,its strict theoretical analysis is not found.In this paper,the error analysis and numerical test of FBSDEs’ explicit differential multistep method are presented.