The maximum principle for partially observed optimal control of forward-backward stochastic systems with random jumps

This paper studies the problem of partially observed optimal control for forward-backward stochastic systems which are driven both by Brownian motions and an independent Poisson random measure. Combining forward-backward stochastic differential equation theory with certain classical convex variational techniques, the necessary maximum principle is proved for the partially observed optimal control, where the control domain is a nonempty convex set. Under certain convexity assumptions, the author also gives the sufficient conditions of an optimal control for the aforementioned optimal optimal problem. To illustrate the theoretical result, the author also works out an example of partial information linear-quadratic optimal control, and finds an explicit expression of the corresponding optimal control by applying the necessary and sufficient maximum principle.     

A Mean-Field Optimal Control for Fully Coupled Forward-Backward Stochastic Control Systems with Lévy Processes

This paper is concerned with a class of mean-field type stochastic optimal control systems, which are governed by fully coupled mean-field forward-backward stochastic differential equations with Teugels martingales associated to Lévy processes. In these systems, the coefficients contain not only the state processes but also their marginal distribution, and the cost function is of mean-field type as well. The necessary and sufficient conditions for such optimal problems are obtained. Furthermore, the applications to the linear quadratic stochastic optimization control problem are investigated.

     

A Maximum Principle for Fully Coupled Forward-Backward Stochastic Control System Driven by Lvy Process with Terminal State Constraints

This paper is concerned with a fully coupled forward-backward stochastic optimal control problem where the controlled system is driven by L′evy 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.     

Stochastic maximum principle for mixed regular-singular control problems of forward-backward systems

This paper considers a stochastic optimal control problem of a forward-backward system with regular-singular controls where the set of regular controls is not necessarily convex and the regular control enters the diffusion coefficient. This control problem is difficult to solve with the classical method of spike variation. The authors use the approach of relaxed controls to establish maximum principle for this stochastic optimal control problem. Sufficient optimality conditions are also investigated.     

MAXIMUM PRINCIPLE FOR OPTIMAL CONTROLPROBLEM OF FULLY COUPLEDFORWARD-BACKWARD STOCHASTIC SYSTEMS

1.IntroductionLet(fi,F,P)beaprobabilityspaceand{Bt}tZobead-dimensionalBrownianmotioninthisspace.Let{R}tZobethenaturalfiltrationofthisBrownianmotion.Weconsiderthefollowingfullycoupledforward-backwardstochasticsystems:where(x,y,z)takesvaluesinR"xacxRTnxd.LetUbeanonemptyconvexsubsetofR',Anelementofadiscalledanadmissiblecontrol.Wecandefinethefollowingcostfunction:TheoptimalcontrolproblemistominimizethecostfunctionJ(v(.))overadmissiblecontrols.Anadmissiblecontrolu(.)iscalledanoptimalcontrol…     

Controlled mean-field backward stochastic differential equations with jumps involving the value function

This paper discusses mean-field backward stochastic differential equations (mean-field BSDEs) with jumps and a new type of controlled mean-field BSDEs with jumps, namely mean-field BSDEs with jumps strongly coupled with the value function of the associated control problem. The authors first prove the existence and the uniqueness as well as a comparison theorem for the above two types of BSDEs. For this the authors use an approximation method. Then, with the help of the notion of stochastic backward semigroups introduced by Peng in 1997, the authors get the dynamic programming principle (DPP) for the value functions. Furthermore, the authors prove that the value function is a viscosity solution of the associated nonlocal Hamilton-Jacobi-Bellman (HJB) integro-partial differential equation, which is unique in an adequate space of continuous functions introduced by Barles, et al. in 1997.     

FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS, LINEAR QUADRATIC STOCHASTIC OPTIMAL CONTROL AND NONZERO SUM DIFFERENTIAL GAMES

In this paper, we use the solutions of forward-backward stochastic differential equations to get the explicit form of the optimal control for linear quadratic stochastic optimal control problem and the open-loop Nash equilibrium point for nonzero sum differential games problem. We also discuss the solvability of the generalized Riccati equation system and give the linear feedback regulator for the optimal control problem using the solution of this kind of Riccati equation system.     

Stochastic maximum principle for optimal control problems of forward-backward delay systems involving impulse controls

This paper is concerned with the optimal control problems of forward-backward delay systems involving impulse controls. The authors establish a stochastic maximum principle for this kind of systems. The most distinguishing features of the proposed problem are that the control variables consist of regular and impulsive controls, both with time delay, and that the domain of regular control is not necessarily convex. The authors obtain the necessary and sufficient conditions for optimal controls, which have potential applications in mathematical finance.     

A maximum principle for general backward stochastic differential equation

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.     

Maximum principle for forward-backward stochastic control system with random jumps and applications to finance

<正> Both necessary and sufficient maximum principles for optimal control of stochastic systemwith random jumps consisting of forward and backward state variables are proved.The control variableis allowed to enter both diffusion and jump coefficients.The result is applied to a mean-varianceportfolio selection mixed with a recursive utility functional optimization problem.Explicit expressionof the optimal portfolio selection strategy is obtained in the state feedback form.     

The Optimal Control of Fully-Coupled Forward-Backward Doubly Stochastic Systems Driven by It?-Lévy Processes

This paper studies the optimal control of a fully-coupled forward-backward doubly stochastic system driven by It?-Lévy processes under partial information. The existence and uniqueness of the solution are obtained for a type of fully-coupled forward-backward doubly stochastic differential equations(FBDSDEs in short). As a necessary condition of the optimal control, the authors get the stochastic maximum principle with the control domain being convex and the control variable being contained in all coefficients. The proposed results are applied to solve the forward-backward doubly stochastic linear quadratic optimal control problem.     

The Stochastic Maximum Principle for a Jump-Diffusion Mean-Field Model Involving Impulse Controls and Applications in Finance

This paper establishes a stochastic maximum principle for a stochastic control of mean-field model which is governed by a Lévy process involving continuous and impulse control. The authors also show the existence and uniqueness of the solution to a jump-diffusion mean-field stochastic differential equation involving impulse control. As for its application, a mean-variance portfolio selection problem has been solved.     

Maximum Principle of Optimal Stochastic Control with Terminal State Constraint and Its Application in Finance

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.     

A Mean-Field Linear-Quadratic Stochastic Stackelberg Differential Game with one Leader and Two Followers

This paper is concerned with a linear-quadratic(LQ) stochastic Stackelberg differential game with one leader and two followers, where the game system is governed by a mean-field stochastic differential equation(MF-SDE). By maximum principle and verification theorem, the open-loop Stackelberg solution is expressed as a feedback form of the state and its mean with the help of three systems of Riccati equations.     

Maximum principle for partially-observed optimal control problems of stochastic delay systems

This paper is concerned with partially-observed optimal control problems for stochastic delay systems. Combining Girsanov’s theorem with a standard variational technique, the authors obtain a maximum principle on the assumption that the system equation contains time delay and the control domain is convex. The related adjoint processes are characterized as solutions to anticipated backward stochastic differential equations in finite-dimensional spaces. Then, the proposed theoretical result is applied to study partially-observed linear-quadratic optimal control problem for stochastic delay system and an explicit observable control variable is given.     

Discrete-time mean-field stochastic <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control

The finite horizon H ₂/H _∞ control problem of mean-field type for discrete-time systems is considered in this paper. Firstly, the authors derive a mean-field stochastic bounded real lemma (SBRL). Secondly, a sufficient condition for the solvability of discrete-time mean-field stochastic linearquadratic (LQ) optimal control is presented. Thirdly, based on SBRL and LQ results, this paper establishes a sufficient condition for the existence of discrete-time stochastic H ₂/H _∞ control of meanfield type via the solvability of coupled matrix-valued equations.     

Backward linear-quadratic stochastic optimal control and nonzero-sum differential game problem with random jumps

This paper studies the existence and uniqueness of solutions of fully coupled forward-backward stochastic differential equations with Brownian motion and random jumps. The result is applied to solve a linear-quadratic optimal control and a nonzero-sum differential game of backward stochastic differential equations. The optimal control and Nash equilibrium point are explicitly derived. Also the solvability of a kind Riccati equations is discussed. All these results develop those of Lim, Zhou (2001) and Yu, Ji (2008).     

Optimal Reinsurance and Investment Strategies Under Mean-Variance Criteria: Partial and Full Information

This paper is concerned with an optimal reinsurance and investment problem for an insurance firm under the criterion of mean-variance. The driving Brownian motion and the rate in return of the risky asset price dynamic equation cannot be directly observed. And the short-selling of stocks is prohibited. The problem is formulated as a stochastic linear-quadratic control problem where the control variables are constrained. Based on the separation principle and stochastic filtering theory, the partial information problem is solved. Efficient strategies and efficient frontier are presented in closed forms via solutions to two extended stochastic Riccati equations. As a comparison, the efficient strategies and efficient frontier are given by the viscosity solution to the HJB equation in the full information case. Some numerical illustrations are also provided.

     

基于限价订单簿的最优高频做市商策略研究

建立具有成交风险和存货风险的价差过程模型,在引入存货惩罚函数的同时将策略的目标确定为效用最大化.将策略求解的过程看成是随机最优控制问题,并通过动态规划求解,离散模型框架下采用有限差分的方法对每个时间点不同存货及市场价差下的下单策略进行求解.该策略满足了模型定义之初对于成交强度,市场价差及存货量对下单行为影响的假设,而策略的实证及可靠性检验进一步表明了该策略具有较为稳定的收益.     

Stochastic differential equations and stochastic linear quadratic optimal control problem with Lévy processes

In this paper, the authors first study two kinds of stochastic differential equations (SDEs) with Lévy 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 Lévy processes, the authors proceed to study a stochastic linear quadratic (LQ) optimal control problem with a Lévy 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. This work was supported by the National Basic Research Program of China (973 Program) under Grant No. 2007CB814904, the Natural Science Foundation of China under Grant No. 10671112 and Shandong Province under Grant No. Z2006A01, and Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20060422018.