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.     

A varying terminal time structure for stochastic optimal control under constrained condition

In this study, we propose a varying terminal time structure for the optimal control problem under state constraints, in which the terminal time follows the varying of the control via the constrained condition. Focusing on this new optimal control problem, we investigate a novel stochastic maximum principle, which differs from the traditional optimal control problem under state constraints. The optimal pair of the optimal control model can be verified via this new stochastic maximum principle.     

Maximum principle for forward–backward SDEs with a general cost functional

In this study, we consider a stochastic optimal control problem whose state variables are described by the system of forward and backward stochastic differential equations with a general cost functional which relies on the global terminal condition. In the framework of Fréchet derivatives, we derive the corresponding maximum principle via constructing a series of adjoint equations which need to be solved step by step.     

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.     

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.     

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.     

Near-maximum principle for general recursive utility optimal control problem

In this study, we introduce a general recursive utility optimal control problem driven by a fully nonlinear stochastic differential system. In order to investigate Pontryagin's stochastic maximum principle for this new optimal control problem, we establish the near-optimal control system for the original problem and obtain the related near-maximum principle. Two examples are also provided to illustrate the near-maximum principle.     

Necessary and sufficient conditions of risk‐sensitive optimal control and differential games for stochastic differential delayed equations

In this paper, we consider risk‐sensitive optimal control and differential games for stochastic differential delayed equations driven by Brownian motion. The problems are related to robust stochastic optimization with delay due to the inherent feature of the risk‐sensitive objective functional. For both problems, by using the logarithmic transformation of the associated risk‐neutral problem, the necessary and sufficient conditions for the risk‐sensitive maximum principle are obtained. We show that these conditions are characterized in terms of the variational inequality and the coupled anticipated backward stochastic differential equations (ABSDEs). The coupled ABSDEs consist of the first‐order adjoint equation and an additional scalar ABSDE, where the latter is induced due to the nonsmooth nonlinear transformation of the adjoint process of the associated risk‐neutral problem. For applications, we consider the risk‐sensitive linear‐quadratic control and game problems with delay, and the optimal consumption and production game, for which we obtain explicit optimal solutions.     

Stochastic maximum principle for optimal control problems involving delayed systems

Dear editor, The main objective of this study is to investigate one type of stochastic optimal control problem for a delayed system using the maximum principle ...     

约束随机线性二次最优控制的研究

本文研究线性终端状态约束下不定随机线性二次最优控制问题.首先利用Lagrange Multiplier 定理得到了存在最优线性状态反馈解的必要条件, 而在加强的条件下也得到了最优控制存在的充分条件. 从某种意义上讲, 以往关于无约束随机线性二次最优控制的一些结果可以看成本文主要定理的推论.     

Near‐optimal control for a stochastic SIRS model with imprecise parameters

Parameter values are usually assumed to be precisely known in many epidemic models but they could be imprecise due to various uncertainties. In this paper, we develop a stochastic SIRS model that includes imprecise parameters and white noise, formulate and analyze the near‐optimal control problem for the stochastic model. We obtain priori estimates of the susceptible, infected and recovered populations. Sufficient and necessary conditions for the near optimality of the model are established using Ekeland's principle and a nearly maximum condition on the Hamiltonian function. Numerical simulations are also performed to demonstrate the analytical results and evaluate the influence of imprecise parameters, white noise and treatment control on the dynamics of epidemics.     

带Markov跳的离散时间随机控制系统的最大值原理

本文研究一类同时含有Markov跳过程和乘性噪声的离散时间非线性随机系统的最优控制问题, 给出并证明了相应的最大值原理. 首先, 利用条件期望的平滑性, 通过引入具有适应解的倒向随机差分方程, 给出了带有线性差分方程约束的线性泛函的表示形式, 并利用Riesz定理证明其唯一性. 其次, 对带Markov跳的非线性随机控制系统, 利用针状变分法, 对状态方程进行一阶变分, 获得其变分所满足的线性差分方程. 然后, 在引入Hamilton函数的基础上, 通过一对由倒向随机差分方程刻画的伴随方程, 给出并证明了带有Markov跳的离散时间非线性随机最优控制问题的最大值原理, 并给出该最优控制问题的一个充分条件和相应的Hamilton-Jacobi-Bellman方程. 最后, 通过 一个实际例子说明了所提理论的实用性和可行性.     

Near-optimal control for stochastic recursive problems

It is well documented (e.g. Zhou (1998) [8]) that the near-optimal controls, as the alternative to the “exact” optimal controls, are of great importance for both the theoretical analysis and practical application purposes due to its nice structure and broad-range availability, feasibility as well as flexibility. However, the study of near-optimality on the stochastic recursive problems, to the best of our knowledge, is a totally unexplored area. Thus we aim to fill this gap in this paper. As the theoretical result, a necessary condition as well as a sufficient condition of near-optimality for stochastic recursive problems is derived by using Ekeland’s principle. Moreover, we work out an ε-optimal control example to shed light on the application of the theoretical result. Our work develops that of [8] but in a rather different backward stochastic differential equation (BSDE) context.     

Robust stochastic maximum principle for multi-model worst case optimization

This paper develops a version of the robust maximum principle applied to the minimax Mayer problem formulated for stochastic differential equations with a control-dependent diffusion term. The parametric families of first and second order adjoint stochastic processes are introduced to construct the corresponding Hamiltonian formalism. The Hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic Hamiltonians corresponding to each possible value of the parameter. The cost function is defined on a finite horizon and contains the mathematical expectation of a terminal term. A terminal condition, given by a vector function, is also considered. The optimal control strategies, adapted for available information, for the wide class of multi-model systems given by a stochastic differential equation with parameters from a given finite set are constructed. This problem belongs to the class of minimax stochastic optimization problems. The proof is based on the recent results obtained for deterministic minimax Mayer problem by Boltyanski and Poznyak as well as on the results of Zhou and of Yong and Zhou, obtained for stochastic maximum principle for non-linear stochastic systems with a single-valued parameter. Two illustrative examples, dealing with production planning and reinsurance-dividend management, conclude this study.     

Stochastic minimax optimal control strategy for uncertain quasi-Hamiltonian systems using stochastic maximum principle

A stochastic minimax optimal control strategy for uncertain quasi-Hamiltonian systems is proposed based on the stochastic averaging method, stochastic maximum principle and stochastic differential game theory. First, the partially completed averaged Itô stochastic differential equations are derived from a given system by using the stochastic averaging method for quasi-Hamiltonian systems with uncertain parameters. Then, the stochastic Hamiltonian system for minimax optimal control with a given performance index is established based on the stochastic maximum principle. The worst disturbances are determined by minimizing the Hamiltonian function, and the worst-case optimal controls are obtained by maximizing the minimal Hamiltonian function. The differential equation for adjoint process as a function of system energy is derived from the adjoint equation by using the Itô differential rule. Finally, two examples of controlled uncertain quasi-Hamiltonian systems are worked out to illustrate the application and effectiveness of the proposed control strategy.     

The maximum principle for partially observed optimal control problems of mean-field FBSDEs

This paper is concerned with a partially observed optimal control problem described by mean-field forward and backward stochastic differential equations. Moreover, the control variable enters the diffusion coefficient and the control domain is non-convex. Utilising Girsanov's theorem as well as extended Ekeland's variational principle, a maximum principle is established in the form of Pontryagin's type. As an application, a linear-quadratic control problem is studied in terms of the stochastic filtering.     

Input-to-state stability of stochastic nonlinear fuzzy Cohen–Grossberg neural networks with the event-triggered control

ABSTRACT

In this paper, we investigate a class of stochastic nonlinear fuzzy Cohen–Grossberg neural networks with feedback control and an unknown exogenous disturbance. By using the Lyapunov function, Itô's formula, Dynkin's formula, Comparison principle and stochastic analysis theory, we show that the considered system is input-to-state stable with the help of the designed event-triggered mechanism. Moreover, we also guarantee that the internal execution time intervals of control task will not be arbitrarily small. Finally, some remarks and discussions have been provided to show that our results are meaningful.     

Feasible solution to discrete-time linear quadratic stochastic Stackelberg difference game

This paper investigates the discrete-time linear quadratic (LQ) stochastic Stackelberg game, which has not been thoroughly addressed in previous literature. Firstly, we derive the maximum principle for the stochastic Stackelberg difference game using the variational method, and obtain the necessary and sufficient solvability conditions. However, due to the coupling between the two players and the presence of stochastic noise, obtaining explicit optimal leader and follower's strategies becomes challenging. Therefore, we present a feasible suboptimal control strategy instead. As a result, we derive a feasible suboptimal control strategy. To achieve this, we assume a linear homogeneous relationship to decouple the group of stochastic game forward-backward stochastic differential equations (SG-FBSDEs), which serves as a compromise for obtaining the optimal solution. With this approach, we derive a feasible solution to the stochastic Stackelberg difference game based on the solution to symmetric Riccati equations.     

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.     

Identification and Control of Multivariable Stochastic Feedback Systems

Abstract

This paper considers the problem of simultaneous identification and control of stochastic processes characterized by linear dynamic models with unknown systems parameter coefficients. Stochastic approximation is used to derive consistent identification algorithms for the case in which arbitrary feedback controls are present. These identification methods can also be used for determining the order of the system, if the latter is unknown, as well as the exact canonical structure for the multivariable case.

An approximation to the optimal control solution is obtained by explicitly separating the functions of identification and control, and asymptotic convergence to a stochastic optimal controller is attained without on-line structural modification.