Mixed Optimal Control Problem Of Forward-Backward Stochastic System And Its Applications In Economics

This dissertation is devoted to the study of the mixed optimal control problem of forward-backward stochastic system and its applications in economics.It is mainly on the optimal control problem of backward stochastic mixed control system,the non-zero sum mixed differential game of backward stochastic differential equation,and the mixed optimal control problem of forward-backward stochastic system and its applications in economics.The main contributions are as follows:For the first time,we obtain the neces-sary and sufficient conditions for the optimal control problem of backward stochastic mixed control system.Moreover,the feedback optimal control with mean-field term is obtained.Furthermore,the optimal state satisfies a mean-field backward stochas-tic differential equation,and we give the main reason why mean-field term appears in the optimal control and the corresponding optimal state equation.We establish a necessary condition and an Arrow's sufficient condition for open-loop equilibrium point,and design the unique feedback optimal control for the forward-backward s-tochastic mixed control system based on decoupled Riccati equations and differential equations.We also apply this result to a portfolio problem.Based on the relationship between nonlinear expectation,convex risk measure and backward stochastic differ-ential equation,a class of security investment and cyber insurance problem can be equivalently transformed into a type of forward-backward stochastic mixed optimal control problem.We derive the necessary and sufficient conditions for the mixed optimal control problem of forward-backward stochastic differential equation.For mean-field backward stochastic differential equation with non-Lipschitz coefficient,a Picard iteration with Bihari inequality is used to deduce the existence and unique-ness,this result is shown to provide a theoretical basis for studying the mixed optimal control problem driven by mean-field backward stochastic differential equation.The main innovations are as follows:It is first that the theoretical technique is proposed to solve a class of backward stochastic optimal control with mixed deter-ministic controller and random controller.For linear-quadratic case,we obtain the explicit form of optimal control,and the optimal state satisfies a kind of mean-field backward stochastic differential equation.We also apply this result to solve a kind of product management problem.For the stochastic differential game problem of backward stochastic mixed system,it is first that get the necessary and sufficien-t conditions for the equilibrium point under an Arrow sufficient condition.With the help of newly introduced auxiliary equations and decoupling method,the rela-tionship between the state equation and adjoint process are established.Moreover,we obtain the relationship between equilibrium point and state equation,and apply the theoretical results to solve a kind of home mortgage and wealth management problem.It is first time that apply the theoretical results about forward-backward stochastic mixed optimal control to solve a kind of problem of security investment and cyber insuranceThe main contents,results and innovations are listed as follows in the order of chapters:1.We consider an optimal control problem driven by backward stochastic mixed system and its application in linear-quadratic control and in a product man-agement problem.Firstly,by using the convex variational method and the stochastic maximum principle,the necessary condition and sufficient condition for backward stochastic mixed optimal control in the nonlinear case is derived.Next,consider a linear-quadratic case based on the nonlinear case.By the stochastic maximum principle,a candidate optimal control is obtained,and its optimality can be verified by sufficient condition.Using the decoupling technique,we can obtain the feed-back optimal control.We also get that the optimal control is expressed by the mean of state equation and the state equation,and the optimal state equation satisfies a mean-field backward stochastic differential equation.Finally,from the case with on-ly stochastic control and the case with only deterministic control,we get the main reason why mean-field term appears in the optimal control and the corresponding optimal state equation.That is,when the state equation only contains determinis-tic control,but its cost functional still contains both stochastic control process and deterministic control.Furthermore,the above results are applied to solve a kind of problem about production,labor strategy and the intervention of government policy.The main innovation is that the optimal control is explicitly expressed by the solu-tion of mean-field backward stochastic differential equation,which naturally arises from the study of mixed optimal control driven by backward stochastic differential equation without mean-field term.2.We investigate a kind of non-zero sum mixed differential game of backward stochastic differential equation.Different from the case only contains stochastic control,using convex variational method,the maximum principle of backward s-tochastic mixed differential game problem is established.Moreover,the sufficient condition for the equilibrium point is obtained by verifying Arrow condition.We use a numerical example to show that the necessity of convexity in Arrow condition.For the linear-quadratic case,we obtain a non-classical forward-backward stochastic differential equation by decoupling technique.Furthermore,we obtain the existence and uniqueness of this non-classical forward-backward stochastic differential equa-tion by introducing two auxiliary equations under certain conditions.We obtain the relationship between the equilibrium point and state equation by the decoupling technique,the solution of Riccati equation and differential equation.It should be noted that the non-zero sum mixed differential game problem of backward stochas-tic differential equation are solved in this paper,which lays a theoretical foundation for the study of the differential game problems under the influence of government s-trategies,optimal portfolio game problems,and time-inconsistent mean-field system game problems.3.The forward-backward stochastic mixed optimal control problem is solved in the theory of stochastic optimal control and convexity of function.For partially coupled controlled forward-backward stochastic mixed system,we obtain the nec-essary and sufficient conditions for the existence of optimal control by introducing two new adjoint equations and the stochastic maximum principle.Moreover,we derive the explicit optimal control in terms of two decoupled Riccati equations and five ordinary differential equations.By decoupling technique and the method of first finding the relationship between the state and the adjoint process and then the rela-tionship between the adjoint process and the state,the feedback optimal control is obtained.The main innovation is that this problem is motivated by a kind of security investment and cyber insurance problem arising from financial mathematics.It is first that research on the theoretical of forward-backward stochastic mixed optimal control problem.Based on the relationship between Peng's nonlinear mathematical expectation,backward stochastic differential equation and convex risk measure,a class of security investment and cyber insurance problem is equivalently converted to an optimal control problem for forward-backward stochastic mixed system.These results perfect and improve stochastic mixed control theory.4.For the stochastic mixed optimal control problem and stochastic mixed d-ifferential game problem driven by backward stochastic differential equation,both problems can produce a kind of mean-field backward stochastic differential equa-tion.Inspired by the above phenomenon,we study a new existence and uniqueness result of solution for mean-field backward stochastic differential equation.Using Picard iteration and Ito's formula,the estimation of the sequence of state equation is obtained.Furthermore,by constructing two monotonic function columns,the exis-tence of the solution is derived.Combining Bihari inequality,we get the uniqueness of mean-field backward stochastic differential equation.The main innovation is that we get a new existence and uniqueness result of a solution for mean-field backward stochastic differential equation,where its coefficient is satisfies non-Lipschitz con-dition.It lays a theoretical foundation for the study of forward-backward stochastic mixed optimal control problem with non-Lipschitz condition.