Maximum Principle For Stochastic Controlled Systems Driven By Fractional Brownian Motions

The research of stochastic optimal control theory aims at seeking method and theory to solve the stochastic optimal control problems,including the necessary and sufficient conditions of optimal control,the existence and uniqueness of solutions to backward(forward)stochastic differential equations,and the regularity theory of so-lutions to HJB function and so on.Optimal control theory dated from the end of the 1950s,the landmark was the maximum principle put forward by one of mathmatician-s Pontryagin from the Soviet Union,afterwards,principle of optimality by Bellman and filtering theory by Kalman.There are many inescapable stochastic factors in real life.Therefore,it is a significant subject that scholars would research and establish stochastic optimal control theory.In fact,the method of stochastic optimal control theory have been applied to many fields such as automatic control cybernation,signal processing,aerospace technology,mechanical engineering,finance engineering and so on.To study the system of stochastic optimal control problem quantificationally,we should establish the mathematical model with random inputs or stochastic disturbance.However,in the actual operation,many noises can be estimated by white noise process.Therefore,when the system inputs or interferences are other processes,we approximate by white noise.In this way,effective mathematical methods can be applied as well as significant errors are not caused during the process of treatment.In the fields of s-tochastic analysis,Brownian motion can be described as integral form of Gauss white noise.As the basic tools to a depiction of a series of complex processes,there are more perfect theoretical system and research value.Therefore,in deal with practical prob-lems,the stochastic optimal control problem of systems driven by Brownian motions has been studied by many scholars which obtain many abundant research results.Fractional Brownian motion is neither a Markov process nor a semi-martingale,which has two important properties of self-similar and a long range dependence.The two properties are the inherent character of many natural phenomena and social phe-nomena.Therefore,the model of fractional Brownian motion is one of the most widely used models.In fact,the research on the fractional Brownian motion has been success-fully applied to hydrogeology,meteorology,signal process,data flow analysis,finance and so on.Stochastic optimal control problem of systems driven by fractional Browni-an motion has become the current research hotspots in the field of stochastic analysis.In the study of such problems,the solutions of the backward stochastic differential e-quation as the adjoint equation of state equation is the key of the stochastic maximum principle.Backward stochastic differential equations have obtained high international popu-larity in stochastic analysis,stochastic control and financial mathematics.The forward stochastic differential equation is concerned with how to recognize an objective stochas-tic existent process.Backward stochastic differential equations are mainly concerned with how to achieve the expected goal and make a system in a random environmen-t.Although the history of the theory of backward stochastic differential equations is relatively short,the progress has been very fast.In addition to the interesting mathematical properties of the theory itself,authors have also found many impor-tant application prospects.Duffie and Epstein,the famous economists,find that can be used to describe consumption preference under economic environment uncertainty.Peng obtained nonlinear Feynman-Kac formula through backward stochastic differen-tial equation.Ei Karoui and Quenez find many theoretical price of many important derives securities in the financial market which can be solved by backward stochastic differential equations.In 2013,Han,Hu and Song in[1]have obtained the maximum principle for general controlled systems driven by fractional Brownian motions with H G(1/2,1)in the convex controlled domain.They get a backward stochastic differential equation which was driven by the fractional Brownian motions and the underlying Brownian motions.In order to improve the stochastic maximum principle,we are interested in the existence and uniqueness of solution of generalized backward stochastic differential equation driven by both fractional Brownian motion and standard Brownian motion.At the same time,we will continue studying the systems but the control domain U is not convex necessary.In this article,we will consider the following problems.Problem 1 When Hurst parameter H ?(1/2,1);we consider the existence and uniqueness of solutions of generalized backward stochastic differential equation driven by both fractional Brownian motion and standard Brownian motion.Problem 2 We consider the general controlled systems driven by fractional Brow-nian motions with H ?(1/2,1).In chapter 3,we consider the following nonlinear backward stochastic differential equations(BSDEs)and aim at proving the local existence and uniquess to the solutions.-dYt = f(t,Yt,Zt)dt + g(t,Yt,Zt)dBtH-ZtdWt,YT = ?,where T ?(0,+?)is a finite undetermined time.In chapter 4,consider the following stochastic control systems:For any u(·)?Uad([0,T]),cost functional is defined byNotice that it is distinguish from[1]that the control domain need not be convex in this chapter.Then,the stochastic optimal control problems can be described.Find the necessary condition of admissible control u*(·)? Uad([0,T]),satisfying the state function,such that the cost functional take the minimum value.That is,(?).