The state equations of stochastic control problems,which are controlled stochastic differential equations,are proposed to be discretized by the weak midpoint rule and predictor-corrector methods for the Markov chain a...The state equations of stochastic control problems,which are controlled stochastic differential equations,are proposed to be discretized by the weak midpoint rule and predictor-corrector methods for the Markov chain approximation approach. Local consistency of the methods are proved.Numerical tests on a simplified Merton's portfolio model show better simulation to feedback control rules by these two methods, as compared with the weak Euler-Maruyama discretisation used by Krawczyk.This suggests a new approach of improving accuracy of approximating Markov chains for stochastic control problems.展开更多
This paper discusses the H∞ control problem for a class of linear stochastic systems driven by both Brownian motion and Poisson jumps. The authors give the basic theory about stabilities for such systems, including i...This paper discusses the H∞ control problem for a class of linear stochastic systems driven by both Brownian motion and Poisson jumps. The authors give the basic theory about stabilities for such systems, including internal stability and external stability, which enables to prove the bounded real lemma for the systems. By means of Riccati equations, infinite horizon linear stochastic state-feedback H∞ control design is also extended to such systems.展开更多
The authors discuss one type of general forward-backward stochastic differential equations (FBSDEs) with Ito's stochastic delayed equations as the forward equations and anticipated backward stochastic differential...The authors discuss one type of general forward-backward stochastic differential equations (FBSDEs) with Ito's stochastic delayed equations as the forward equations and anticipated backward stochastic differential equations as the backward equations.The existence and uniqueness results of the general FBSDEs are obtained.In the framework of the general FBSDEs in this paper,the explicit form of the optimal control for linear-quadratic stochastic optimal control problem with delay and the Nash equilibrium point for nonzero sum differential games problem with delay are obtained.展开更多
This paper is concerned with a fully coupled forward-backward stochastic optimal control problem where the controlled system is driven by Levy process, while the forward state is constrained in a convex set at the ter...This paper is concerned with a fully coupled forward-backward stochastic optimal control problem where the controlled system is driven by Levy process, while the forward state is constrained in a convex set at the terminal time. The authors use an equivalent backward formulation to deal with the terminal state constraint, and then obtain a stochastic maximum principle by Ekeland's variational principle. Finally, the result is applied to the utility optimization problem in a financial market.展开更多
In this paper, tile authors first study two kinds of stochastic differential equations (SDEs) with Levy processes as noise source. Based on the existence and uniqueness of the solutions of these SDEs and multi-dimen...In this paper, tile authors first study two kinds of stochastic differential equations (SDEs) with Levy processes as noise source. Based on the existence and uniqueness of the solutions of these SDEs and multi-dimensional backward stochastic differential equations (BSDEs) driven by Levy pro- cesses, the authors proceed to study a stochastic linear quadratic (LQ) optimal control problem with a Levy process, where the cost weighting matrices of the state and control are allowed to be indefinite. One kind of new stochastic Riccati equation that involves equality and inequality constraints is derived from the idea of square completion and its solvability is proved to be sufficient for the well-posedness and the existence of optimal control which can be of either state feedback or open-loop form of the LQ problems. Moreover, the authors obtain the existence and uniqueness of the solution to the Riccati equation for some special cases. Finally, two examples are presented to illustrate these theoretical results.展开更多
This paper considers the optimal control problem for a general stochastic system with general terminal state constraint. Both the drift and the diffusion coefficients can contain the control variable and the state con...This paper considers the optimal control problem for a general stochastic system with general terminal state constraint. Both the drift and the diffusion coefficients can contain the control variable and the state constraint here is of non-functional type. The author puts forward two ways to understand the target set and the variation set. Then under two kinds of finite-codimensional conditions, the stochastic maximum principles are established, respectively. The main results are proved in two different ways. For the former, separating hyperplane method is used; for the latter, Ekeland's variational principle is applied. At last, the author takes the mean-variance portfolio selection with the box-constraint on strategies as an example to show the application in finance.展开更多
We discuss the stochastic linear-quadratic(LQ) optimal control problem with Poisson processes under the indefinite case. Based on the wellposedness of the LQ problem, the main idea is expressed by the definition of re...We discuss the stochastic linear-quadratic(LQ) optimal control problem with Poisson processes under the indefinite case. Based on the wellposedness of the LQ problem, the main idea is expressed by the definition of relax compensator that extends the stochastic Hamiltonian system and stochastic Riccati equation with Poisson processes(SREP) from the positive definite case to the indefinite case. We mainly study the existence and uniqueness of the solution for the stochastic Hamiltonian system and obtain the optimal control with open-loop form. Then, we further investigate the existence and uniqueness of the solution for SREP in some special case and obtain the optimal control in close-loop form.展开更多
This paper deals with a constrained stochastic linear-quadratic(LQ for short)optimal control problem where the control is constrained in a closed cone. The state process is governed by a controlled SDE with random c...This paper deals with a constrained stochastic linear-quadratic(LQ for short)optimal control problem where the control is constrained in a closed cone. The state process is governed by a controlled SDE with random coefficients. Moreover, there is a random jump of the state process. In mathematical finance, the random jump often represents the default of a counter party. Thanks to the Ito-Tanaka formula, optimal control and optimal value can be obtained by solutions of a system of backward stochastic differential equations(BSDEs for short). The solvability of the BSDEs is obtained by solving a recursive system of BSDEs driven by the Brownian motions. The author also applies the result to the mean variance portfolio selection problem in which the stock price can be affected by the default of a counterparty.展开更多
In this paper,the authors consider a stochastic control problem where the system is governed by a general backward stochastic differential equation.The control domain need not be convex,and the diffusion coefficient c...In this paper,the authors consider a stochastic control problem where the system is governed by a general backward stochastic differential equation.The control domain need not be convex,and the diffusion coefficient can contain a control variable.The authors obtain a stochastic maximum principle for the optimal control of this problem by virtue of the second-order duality method.展开更多
We study a stochastic control system involving both a standard and a fractional Brownian motion with Hurst parameter less than 1/2.We apply an anticipative Girsanov transformation to transform the system into another ...We study a stochastic control system involving both a standard and a fractional Brownian motion with Hurst parameter less than 1/2.We apply an anticipative Girsanov transformation to transform the system into another one,driven only by the standard Brownian motion with coefficients depending on both the fractional Brownian motion and the standard Brownian motion.We derive a maximum principle and the associated stochastic variational inequality,which both are generalizations of the classical case.展开更多
This paper is concerned with the mixed H_2/H_∞ control problem for a new class of stochastic systems with exogenous disturbance signal.The most distinguishing feature,compared with the existing literatures,is that th...This paper is concerned with the mixed H_2/H_∞ control problem for a new class of stochastic systems with exogenous disturbance signal.The most distinguishing feature,compared with the existing literatures,is that the systems are described by linear backward stochastic differential equations(BSDEs).The solution to this problem is obtained completely and explicitly by using an approach which is based primarily on the completion-of-squares technique.Two equivalent expressions for the H_2/H_∞ control are presented.Contrary to forward deterministic and stochastic cases,the solution to the backward stochastic H_2/H_∞ control is no longer feedback of the current state;rather,it is feedback of the entire history of the state.展开更多
This paper considers the problem of partially observed optimal control for forward-backward stochastic systems driven by Brownian motions and an independent Poisson random measure with a feature that the cost function...This paper considers the problem of partially observed optimal control for forward-backward stochastic systems driven by Brownian motions and an independent Poisson random measure with a feature that the cost functional is of mean-field type. When the coefficients of the system and the objective performance functionals are allowed to be random, possibly non-Markovian, Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed. The authors also investigate the mean-field type optimal control problem for the system driven by mean-field type forward-backward stochastic differential equations(FBSDEs in short) with jumps, where the coefficients contain not only the state process but also its expectation under partially observed information. The maximum principle is established using convex variational technique. An example is given to illustrate the obtained results.展开更多
This paper is concerned with stochastic H_2/H_∞ control problem for Poisson jump-diffusion systems with(x, u, v)-dependent noise, which are driven by Brownian motion and Poisson random jumps. A stochastic bounded rea...This paper is concerned with stochastic H_2/H_∞ control problem for Poisson jump-diffusion systems with(x, u, v)-dependent noise, which are driven by Brownian motion and Poisson random jumps. A stochastic bounded real lemma(SBRL for short) for Poisson jump-diffusion systems is firstly established, which stands out on its own as a very interesting theoretical problem. Further, sufficient and necessary conditions for the existence of a state feedback H_2/H_∞ control are given based on four coupled matrix Riccati equations. Finally, a discrete approximation algorithm and an example are presented.展开更多
基金Supported by the Academic Discipline Program,211 Project for Shanghai University of Finance and Economics(the 3rd phase)the Cultivation Fund of the Key Scientific and Technical Innovation Project,Ministry of Education of China (708040)
基金supported by the China Postdoctoral Science Foundation (No.20080430402).
文摘The state equations of stochastic control problems,which are controlled stochastic differential equations,are proposed to be discretized by the weak midpoint rule and predictor-corrector methods for the Markov chain approximation approach. Local consistency of the methods are proved.Numerical tests on a simplified Merton's portfolio model show better simulation to feedback control rules by these two methods, as compared with the weak Euler-Maruyama discretisation used by Krawczyk.This suggests a new approach of improving accuracy of approximating Markov chains for stochastic control problems.
基金supported by the National Natural Science Foundation of China under Grant Nos.60874032 and 70971079
文摘This paper discusses the H∞ control problem for a class of linear stochastic systems driven by both Brownian motion and Poisson jumps. The authors give the basic theory about stabilities for such systems, including internal stability and external stability, which enables to prove the bounded real lemma for the systems. By means of Riccati equations, infinite horizon linear stochastic state-feedback H∞ control design is also extended to such systems.
基金Project supported by the 973 National Basic Research Program of China (No. 2007CB814904)the National Natural Science Foundations of China (No. 10921101)+2 种基金the Shandong Provincial Natural Science Foundation of China (No. 2008BS01024)the Science Fund for Distinguished Young Scholars of Shandong Province (No. JQ200801)the Shandong University Science Fund for Distinguished Young Scholars(No. 2009JQ004)
文摘The authors discuss one type of general forward-backward stochastic differential equations (FBSDEs) with Ito's stochastic delayed equations as the forward equations and anticipated backward stochastic differential equations as the backward equations.The existence and uniqueness results of the general FBSDEs are obtained.In the framework of the general FBSDEs in this paper,the explicit form of the optimal control for linear-quadratic stochastic optimal control problem with delay and the Nash equilibrium point for nonzero sum differential games problem with delay are obtained.
基金supported by the National Science Fundation of China under Grant No.11271007the National Social Science Fund Project of China under Grant No.17BGL058Humanity and Social Science Research Foundation of Ministry of Education of China under Grant No.15YJA790051
文摘This paper is concerned with a fully coupled forward-backward stochastic optimal control problem where the controlled system is driven by Levy process, while the forward state is constrained in a convex set at the terminal time. The authors use an equivalent backward formulation to deal with the terminal state constraint, and then obtain a stochastic maximum principle by Ekeland's variational principle. Finally, the result is applied to the utility optimization problem in a financial market.
基金This work was supported by the National Basic Research Program of China (973 Program) under Grant No. 2007CB814904the Natural Science Foundation of China under Grant No. 10671112+1 种基金Shandong Province under Grant No. Z2006A01Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20060422018
文摘In this paper, tile authors first study two kinds of stochastic differential equations (SDEs) with Levy processes as noise source. Based on the existence and uniqueness of the solutions of these SDEs and multi-dimensional backward stochastic differential equations (BSDEs) driven by Levy pro- cesses, the authors proceed to study a stochastic linear quadratic (LQ) optimal control problem with a Levy process, where the cost weighting matrices of the state and control are allowed to be indefinite. One kind of new stochastic Riccati equation that involves equality and inequality constraints is derived from the idea of square completion and its solvability is proved to be sufficient for the well-posedness and the existence of optimal control which can be of either state feedback or open-loop form of the LQ problems. Moreover, the authors obtain the existence and uniqueness of the solution to the Riccati equation for some special cases. Finally, two examples are presented to illustrate these theoretical results.
基金supported by the National Natural Science Foundation of China under Grant No.11171076Science and Technology Commission,Shanghai Municipality under Grant No.14XD1400400
文摘This paper considers the optimal control problem for a general stochastic system with general terminal state constraint. Both the drift and the diffusion coefficients can contain the control variable and the state constraint here is of non-functional type. The author puts forward two ways to understand the target set and the variation set. Then under two kinds of finite-codimensional conditions, the stochastic maximum principles are established, respectively. The main results are proved in two different ways. For the former, separating hyperplane method is used; for the latter, Ekeland's variational principle is applied. At last, the author takes the mean-variance portfolio selection with the box-constraint on strategies as an example to show the application in finance.
基金supported by National Natural Science Foundation of China (Grant Nos. 61573217,11471192 and 11626142)the National High-Level Personnel of Special Support Program,the Chang Jiang Scholar Program of Chinese Education Ministry+2 种基金the Natural Science Foundation of Shandong Province (Grant Nos. JQ201401 and ZR2016AB08)the Colleges and Universities Science and Technology Plan Project of Shandong Province (Grant No. J16LI55)the Fostering Project of Dominant Discipline and Talent Team of Shandong University of Finance and Economics
文摘We discuss the stochastic linear-quadratic(LQ) optimal control problem with Poisson processes under the indefinite case. Based on the wellposedness of the LQ problem, the main idea is expressed by the definition of relax compensator that extends the stochastic Hamiltonian system and stochastic Riccati equation with Poisson processes(SREP) from the positive definite case to the indefinite case. We mainly study the existence and uniqueness of the solution for the stochastic Hamiltonian system and obtain the optimal control with open-loop form. Then, we further investigate the existence and uniqueness of the solution for SREP in some special case and obtain the optimal control in close-loop form.
基金supported by the National Natural Science Foundation of China(Nos.10325101,11171076)the Shanghai Outstanding Academic Leaders Plan(No.14XD1400400)
文摘This paper deals with a constrained stochastic linear-quadratic(LQ for short)optimal control problem where the control is constrained in a closed cone. The state process is governed by a controlled SDE with random coefficients. Moreover, there is a random jump of the state process. In mathematical finance, the random jump often represents the default of a counter party. Thanks to the Ito-Tanaka formula, optimal control and optimal value can be obtained by solutions of a system of backward stochastic differential equations(BSDEs for short). The solvability of the BSDEs is obtained by solving a recursive system of BSDEs driven by the Brownian motions. The author also applies the result to the mean variance portfolio selection problem in which the stock price can be affected by the default of a counterparty.
文摘In this paper,the authors consider a stochastic control problem where the system is governed by a general backward stochastic differential equation.The control domain need not be convex,and the diffusion coefficient can contain a control variable.The authors obtain a stochastic maximum principle for the optimal control of this problem by virtue of the second-order duality method.
基金supported by National Natural Science Foundation of China(Grant No11301560)
文摘We study a stochastic control system involving both a standard and a fractional Brownian motion with Hurst parameter less than 1/2.We apply an anticipative Girsanov transformation to transform the system into another one,driven only by the standard Brownian motion with coefficients depending on both the fractional Brownian motion and the standard Brownian motion.We derive a maximum principle and the associated stochastic variational inequality,which both are generalizations of the classical case.
基金supported by the Doctoral Foundation of University of Jinan under Grant No.XBS1213
文摘This paper is concerned with the mixed H_2/H_∞ control problem for a new class of stochastic systems with exogenous disturbance signal.The most distinguishing feature,compared with the existing literatures,is that the systems are described by linear backward stochastic differential equations(BSDEs).The solution to this problem is obtained completely and explicitly by using an approach which is based primarily on the completion-of-squares technique.Two equivalent expressions for the H_2/H_∞ control are presented.Contrary to forward deterministic and stochastic cases,the solution to the backward stochastic H_2/H_∞ control is no longer feedback of the current state;rather,it is feedback of the entire history of the state.
基金supported by the National Natural Science Foundation of China under Grant Nos.11471051 and 11371362the Teaching Mode Reform Project of BUPT under Grant No.BUPT2015JY52+5 种基金supported by the National Natural Science Foundation of China under Grant No.11371029the Natural Science Foundation of Anhui Province under Grant No.1508085JGD10supported by the National Natural Science Foundation of China under Grant No.71373043the National Social Science Foundation of China under Grant No.14AZD121the Scientific Research Project Achievement of UIBE NetworkingCollaboration Center for China’s Multinational Business under Grant No.201502YY003A
文摘This paper considers the problem of partially observed optimal control for forward-backward stochastic systems driven by Brownian motions and an independent Poisson random measure with a feature that the cost functional is of mean-field type. When the coefficients of the system and the objective performance functionals are allowed to be random, possibly non-Markovian, Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed. The authors also investigate the mean-field type optimal control problem for the system driven by mean-field type forward-backward stochastic differential equations(FBSDEs in short) with jumps, where the coefficients contain not only the state process but also its expectation under partially observed information. The maximum principle is established using convex variational technique. An example is given to illustrate the obtained results.
基金supported by the Special Funds of the National Natural Science Foundation of China(No.11426154)
文摘This paper is concerned with stochastic H_2/H_∞ control problem for Poisson jump-diffusion systems with(x, u, v)-dependent noise, which are driven by Brownian motion and Poisson random jumps. A stochastic bounded real lemma(SBRL for short) for Poisson jump-diffusion systems is firstly established, which stands out on its own as a very interesting theoretical problem. Further, sufficient and necessary conditions for the existence of a state feedback H_2/H_∞ control are given based on four coupled matrix Riccati equations. Finally, a discrete approximation algorithm and an example are presented.