Global maximum principle for the forward–backward stochastic optimal control problem with poisson jumps

This paper is concerned with the forward–backward stochastic optimal control problem with Poisson jumps. A necessary condition of optimality in the form of a global maximum principle as well as a sufficient condition of optimality are presented under the assumption that the diffusion and jump coefficients do not contain the control variable, and the control domain need not be convex. The case where there are some state constraints is also discussed. A financial example is discussed to illustrate the application of our result. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

A Partially Observed Nonzero‐Sum Stochastic Differential Game with Delays and its Application to Finance

In this paper, we deal with a new kind of partially observed nonzero‐sum differential game governed by stochastic differential delay equations. One of the special features is that the controlled system and the utility functionals involve both delays in the state variable and the control variables under different observation equations for each player. We obtain a maximum principle and a verification theorem for the game problem by virtue of Girsanov's theorem and the convex variational method. In addition, based on the theoretical results and Malliavin derivative techniques, we solve a production and consumption choice game problem.     

Convexity and convex approximations of discrete-time stochastic control problems with constraints

We investigate constrained optimal control problems for linear stochastic dynamical systems evolving in discrete time. We consider minimization of an expected value cost subject to probabilistic constraints. We study the convexity of a finite-horizon optimization problem in the case where the control policies are affine functions of the disturbance input. We propose an expectation-based method for the convex approximation of probabilistic constraints with polytopic constraint function, and a Linear Matrix Inequality (LMI) method for the convex approximation of probabilistic constraints with ellipsoidal constraint function. Finally, we introduce a class of convex expectation-type constraints that provide tractable approximations of the so-called integrated chance constraints. Performance of these methods and of existing convex approximation methods for probabilistic constraints is compared on a numerical example.     

Optimal control of mean-field backward doubly stochastic systems driven by Itô-Lévy processes

ABSTRACT

In this paper, we introduce a new class of backward doubly stochastic differential equations (in short BDSDE) called mean-field backward doubly stochastic differential equations (in short MFBDSDE) driven by Itô-Lévy processes and study the partial information optimal control problems for backward doubly stochastic systems driven by Itô-Lévy processes of mean-field type, in which the coefficients depend on not only the solution processes but also their expected values. First, using the method of contraction mapping, we prove the existence and uniqueness of the solutions to this kind of MFBDSDE. Then, by the method of convex variation and duality technique, we establish a sufficient and necessary stochastic maximum principle for the stochastic system. Finally, we illustrate our theoretical results by an application to a stochastic linear quadratic optimal control problem of a mean-field backward doubly stochastic system driven by Itô-Lévy processes.     

Suboptimal stochastic H-two/H-infinity design with spectrum constraint

This paper studies the problem of suboptimal state-feedback H-two/H-infinity control of stochastic systems with spectrum constraint. Concretely speaking, a mixed suboptimal H-two/H-infinity controller synthesis together with placing the spectrum into a strip region is considered, which is achieved via solving a convex optimization problem.     

带有随机跳跃干扰的线性二次随机最优控制问题

给出一类布朗运动和泊松过程混合驱动的正倒向随机微分方程解的存在唯一性结果, 应用这一结果研究带有随机跳跃干扰的线性二次随机最优控制问题,并得到最优控制的显式形 式,可以证明最优控制是唯一的.然后,引入和研究一类推广的黎卡提方程系统,讨论该方程系统 的可解性并由该方程的解得到带有随机跳跃干扰的线性二次随机最优控制问题最优的线性反馈.     

The stochastic maximum principle for optimal control problems of delay systems involving continuous and impulse controls

This paper is concerned with a Pontryagin’s maximum principle for stochastic optimal control problems of delay systems with random coefficients involving both continuous and impulse controls. This kind of control problems is motivated by some interesting phenomena arising from economics and finance. We establish a necessary maximum principle and a sufficient verification theorem by virtue of the duality and the convex analysis. To explain the theoretical results, we apply them to a production and consumption choice problem.     

Linear‐Quadratic Optimal Control Problems for Mean‐Field Stochastic Differential Equations with Jumps

In this paper, we study a linear‐quadratic optimal control problem for mean‐field stochastic differential equations driven by a Poisson random martingale measure and a one‐dimensional Brownian motion. Firstly, the existence and uniqueness of the optimal control is obtained by the classic convex variation principle. Secondly, by the duality method, the optimality system, also called the stochastic Hamilton system which turns out to be a linear fully coupled mean‐field forward‐backward stochastic differential equation with jumps, is derived to characterize the optimal control. Thirdly, applying a decoupling technique, we establish the connection between two Riccati equations and the stochastic Hamilton system and then prove the optimal control has a state feedback representation.     

Necessary and sufficient conditions for optimal control of stochastic systems associated with Lévy processes

The paper is concerned with a stochastic optimal control problem where the controlled systems are driven by Teugel’s martingales and an independent multi-dimensional Brownian motion. Necessary and sufficient conditions for an optimal control of the control problem with the control domain being convex are proved by the classical method of convex variation, and the coefficients appearing in the systems are allowed to depend on the control variables. As an application, the linear quadratic stochastic optimal control problem is studied.     

Stochastic maximum principle for SPDEs with noise and control on the boundary

In this paper we prove necessary conditions for optimality of a stochastic control problem for a class of stochastic partial differential equations that is controlled through the boundary. This kind of problem can be interpreted as a stochastic control problem for an evolution system in a Hilbert space. The regularity of the solution of the adjoint equation, that is a backward stochastic equation in infinite dimension, plays a crucial role in the formulation of the maximum principle.     

Global Output‐Feedback Stabilization for Stochastic Nonlinear Systems with Function Control Coefficients

This paper investigates the global output‐feedback stabilization for a class of stochastic nonlinear systems with function control coefficients. Notably, the systems in question possess control coefficients that are functions of output, rather than constants; hence, they are different from the existing literature on stochastic stabilization. To solve the control problem, an appropriate reduced‐order observer is introduced to reconstruct the unmeasured system states before a smooth output‐feedback controller is designed using the backstepping method, which guarantees that the closed‐loop system is globally asymptotically stable in probability. This paper combines the related results in the deterministic and stochastic setting and gives the first treatment on the global output‐feedback stabilization for the stochastic nonlinear systems with function control coefficients. A simulation example is given also to illustrate the effectiveness of the proposed approach.     

Optimal Control Problem for Risk‐Sensitive Mean‐Field Stochastic Delay Differential Equation with Partial Information

This paper deals with the risk‐sensitive control problem for mean‐field stochastic delay differential equations (MF‐SDDEs) with partial information. Firstly, under the assumptions that the control domain is not convex and the value function is non‐smooth, we establish a stochastic maximum principle (SMP). Then, by means of Itô's formula and some continuous dependence, we prove the existence and uniqueness results for another type of MF‐SDDEs. Meanwhile, the verification theorem for the MF‐SDDEs is obtained by using a clever construction of the Hamiltonian function. Finally, based on our verification theorem, a linear‐quadratic system is investigated and the optimal control is also derived by the stochastic filtering technique.     

The maximum principle for stochastic differential systems with general cost functional

In this paper, under the framework of Fréchet derivatives, we study a stochastic optimal control problem driven by a stochastic differential equation with general cost functional. By constructing a series of first-order and second-order adjoint equations, we establish the stochastic maximum principle and get the related Hamilton systems.     

The relaxed optimal control problem for Mean-Field SDEs systems and application

The present study deals with a new approach of optimal control problems where the state equation is a Mean-Field stochastic differential equation, and the set of strict (classical) controls need not be convex and the diffusion coefficient depends on the term control. Our consideration is based on only one adjoint process, and the necessary conditions as well as a sufficient condition for optimality in the form of a relaxed maximum principle are obtained, with application to Linear quadratic stochastic control problem with mean-field type.     

Maximum principle for the stochastic optimal control problem with delay and application

In this paper, we consider an optimal control problem for the stochastic system described by stochastic differential equations with delay. We obtain the maximum principle for the optimal control of this problem by virtue of the duality method and the anticipated backward stochastic differential equations. Our results can be applied to a production and consumption choice problem. The explicit optimal consumption rate is obtained.     

带Poisson跳的线性二次随机微分博弈及其在鲁棒控制中的应用

研究了一类带Poisson跳扩散过程的线性二次随机微分博弈,包括非零和博弈的Nash均衡策略与零和博弈的鞍点均衡策略问题．利用微分博弈的最大值原理,得到Nash均衡策略的存在条件等价于两个交叉耦合的矩阵Riccati方程存在解,鞍点均衡策略的存在条件等价于一个矩阵Riccati方程存在解的结论,并给出了均衡策略的显式表达及最优性能泛函值．最后,将所得结果应用于现代鲁棒控制中的随机H₂/H_∞控制与随机H_∞控制问题,得到了鲁棒控制策略的存在条件及显式表达,并验证所得结果在金融市场投资组合优化问题中的应用．     

Optimal ensemble control of stochastic time-varying linear systems

We consider the optimal guidance of an ensemble of independent, structurally identical, finite-dimensional stochastic linear systems with variation in system parameters between initial and target states of interest by applying a common control function without the use of feedback. Our exploration of such ensemble control systems is motivated by practical control design problems in which variation in system parameters and stochastic effects must be compensated for when state feedback is unavailable, such as in pulse design for nuclear magnetic resonance spectroscopy and imaging. In this paper, we extend the notion of ensemble control to stochastic linear systems with additive noise and jumps, which we model using white Gaussian noise and Poisson counters, respectively, and investigate the optimal steering problem. In our main result, we prove that the minimum norm solution to a Fredholm integral equation of the first kind provides the optimal control that simultaneously minimizes the mean square error (MSE) and the error in the mean of the terminal state. The optimal controls are generated numerically for several example ensemble control problems, and Monte Carlo simulations are used to illustrate their performance. This work has immediate applications to the control of dynamical systems with parameter dispersion or uncertainty that are subject to additive noise, which are of interest in quantum control, neuroscience, and sensorless robotic manipulation.     

Backward Stochastic H2/H∞ Control with Random Jumps

This paper is concerned with H₂/H_∞ control of a new class of stochastic systems. The most distinguishing feature, compared with the existing literature, is that the systems are described by backward stochastic differential equations (BSDEs) with Brownian motion and random jumps. It is shown that the backward stochastic H₂/H_∞ control under consideration is associated with the of the corresponding uncontrolled backward stochastic perturbed system. A necessary and sufficient condition for the existence of a unique solution to the control problem under consideration is derived. The resulting solution is characterized by the solution of an uncontrolled forward backward stochastic differential equation (FBSDE) with Brownian motion and random jumps. When the coefficients are all deterministic, the equivalent linear feedback solution involves a pair of Riccati‐type equations and an uncontrolled BSDE. In addition an uncontrolled forward stochastic differential equation (SDE) is given.     

On optimal control of mean-field stochastic systems driven by Teugels martingales via derivative with respect to measures

ABSTRACT

This paper deals with partial information stochastic optimal control problem for general controlled mean-field systems driven by Teugels martingales associated with some Lévy process having moments of all orders, and an independent Brownian motion. The coefficients of the system depend on the state of the solution process as well as of its probability law and the control variable. We establish a set of necessary conditions in the form of Pontryagin maximum principle for the optimal control. We also give additional conditions, under which the necessary optimality conditions turn out to be sufficient. The proof of our result is based on the derivative with respect to the probability law by applying Lions derivatives and a corresponding Itô formula. As an application, conditional mean-variance portfolio selection problem in incomplete market, where the system is governed by some Gamma process is studied to illustrate our theoretical results.     

Convergence of Solutions and Their Exit Times in Diffusion Models with Jumps

We consider a diffusion model with jumps given by a stochastic differential equation with a finite Poisson measure and coefficients depending on a parameter. It is shown that, in the case of convergence of the coefficients, both the solution and its exit times converge.