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.     

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.     

Approximate controllability of stochastic differential systems driven by a Lévy process

In this paper, we are concerned with the approximate controllability of stochastic differential systems driven by Teugels martingales associated with a Lévy process. We derive the approximate controllability with the coefficients in the system satisfying some non-Lipschitz conditions, which include classic Lipschitz conditions as special cases. The desired result is established by means of standard Picard’s iteration.     

Maximum principle for mean-field jump–diffusion stochastic delay differential equations and its application to finance

This paper investigates a stochastic optimal control problem with delay and of mean-field type, where the controlled state process is governed by a mean-field jump–diffusion stochastic delay differential equation. Two sufficient maximum principles and one necessary maximum principle are established for the underlying system. As an application, a bicriteria mean–variance portfolio selection problem with delay is studied to demonstrate the effectiveness and potential of the proposed techniques. Under certain conditions, explicit expressions are provided for the efficient portfolio and the efficient frontier, which are as elegant as those in the classical mean–variance problem without delays.     

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.     

Continuous-time singular linear–quadratic control: Necessary and sufficient conditions for the existence of regular solutions

The purpose of this paper is to provide a full understanding of the role that the constrained generalized continuous algebraic Riccati equation plays in singular linear–quadratic (LQ) optimal control. Indeed, in spite of the vast literature on LQ problems, only recently a sufficient condition for the existence of a non-impulsive optimal control has for the first time connected this equation with the singular LQ optimal control problem. In this paper, we establish four equivalent conditions providing a complete picture that connects the singular LQ problem with the constrained generalized continuous algebraic Riccati equation and with the geometric properties of the underlying system.     

Optimal control of mean-field jump-diffusion systems with noisy memory

This paper is concerned with an optimal control problem under mean-field jump-diffusion systems with delay and noisy memory. First, we derive necessary and sufficient maximum principles using Malliavin calculus technique. Meanwhile, we introduce a new mean-field backward stochastic differential equation as the adjoint equation which involves not only partial derivatives of the Hamiltonian function but also their Malliavin derivatives. Then, applying a reduction of the noisy memory dynamics to a two-dimensional discrete delay optimal control problem, we establish the second set of necessary and sufficient maximum principles under partial information. Moreover, a natural link between the above two approaches is established via the adjoint equations. Finally, we apply our theoretical results to study a mean-field linear-quadratic optimal control problem.     

Discrete time mean-field stochastic linear-quadratic optimal control problems

This paper firstly presents necessary and sufficient conditions for the solvability of discrete time, mean-field, stochastic linear-quadratic optimal control problems. Secondly, the optimal control within a class of linear feedback controls is investigated using a matrix dynamical optimization method. Thirdly, by introducing several sequences of bounded linear operators, the problem is formulated as an operator stochastic linear-quadratic optimal control problem. By the kernel-range decomposition representation of the expectation operator and its pseudo-inverse, the optimal control is derived using solutions to two algebraic Riccati difference equations. Finally, by completing the square, the two Riccati equations and the optimal control are also obtained.     

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.     

Optimal control for stochastic linear quadratic singular neuro Takagi–Sugeno fuzzy system with singular cost using genetic programming

In this paper, optimal control for stochastic linear quadratic singular neuro Takagi–Sugeno (T-S) fuzzy system with singular cost is obtained using genetic programming(GP). To obtain the optimal control, the solution of matrix Riccati differential equation (MRDE) is computed by solving differential algebraic equation (DAE) using a novel and nontraditional GP approach. The obtained solution in this method is equivalent or very close to the exact solution of the problem. Accuracy of the solution computed by GP approach to the problem is qualitatively better. The solution of this novel method is compared with the traditional Runge–Kutta (RK) method. A numerical example is presented to illustrate the proposed method.     

广义连续随机非线性系统的递推算法

讨论了广义连续随机非线性系统的最优递推问题,利用矩阵的奇异值分解理论,给出了广义连续随机非线性系统的奇异值标准形式.基于标准形式,在两种情况下将系统分解成两个子系统,通过对子系统状态估计的研究,得到了该系统的最优递推算法.     

Wonham Filtering by Observations with Multiplicative Noises

We solve the optimal filtering problem for states of a homogeneous finite-state Markov jump process by indirect observations in the presence of Wiener noise. The key feature of this problem is that the noise intensities in observations depend on the unobserved state. The filtering estimate is represented as a solution to some stochastic system with continuous and purely discontinuous martingales in the right-hand side. We discuss the theoretical results and present a numerical example that illustrates the properties of the obtained estimates.     

Stochastic maximum principle in the mean-field controls

In Buckdahn, Djehiche, Li, and Peng (2009), the authors obtained mean-field Backward Stochastic Differential Equations (BSDEs) in a natural way as a limit of some highly dimensional system of forward and backward SDEs, corresponding to a great number of “particles” (or “agents”). The objective of the present paper is to deepen the investigation of such mean-field BSDEs by studying their stochastic maximum principle. This paper studies the stochastic maximum principle (SMP) for mean-field controls, which is different from the classical ones. This paper deduces an SMP in integral form, and it also gets, under additional assumptions, necessary conditions as well as sufficient conditions for the optimality of a control. As an application, this paper studies a linear quadratic stochastic control problem of mean-field type.     

Optimal control for stochastic linear quadratic singular Takagi–Sugeno fuzzy delay system using genetic programming

In this paper, optimal control for stochastic linear singular Takagi–Sugeno (T–S) fuzzy delay system with quadratic performance is obtained using genetic programming (GP). To obtain the optimal control, the solution of matrix Riccati differential equation (MRDE) is computed by solving differential algebraic equation (DAE) using a novel and nontraditional GP approach. The GP solution is equivalent or very close to the exact solution of the problem. Accuracy of the GP solution to the problem is qualitatively better. The solution of this novel method is compared with the traditional Runge Kutta (RK) method. An illustrative numerical example is presented for the proposed method.     

Optimal control for multi-stage and continuous-time linear singular systems

In this paper, optimal control problems for multi-stage and continuous-time linear singular systems are both considered. The singular systems are assumed to be regular and impulse-free. First, a recurrence equation is derived according to Bellman's principle of optimality in dynamic programming. Then, by applying the recurrence equation, bang-bang optimal controls for the control problems with linear objective functions subject to two types of multi-stage singular systems are obtained. Second, employing the principle of optimality, a equation of optimality for settling the optimal control problem subject to a class of continuous-time singular systems is proposed. The optimal control problem may become simpler through solving this equation of optimality. Two numerical examples and a dynamic input–output model are presented to show the effectiveness of the results obtained.     

Nonlinear–nonquadratic optimal and inverse optimal control for stochastic dynamical systems

In this paper, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for nonlinear stochastic dynamical systems. Specifically, we provide a simplified and tutorial framework for stochastic optimal control and focus on connections between stochastic Lyapunov theory and stochastic Hamilton–Jacobi–Bellman theory. In particular, we show that asymptotic stability in probability of the closed‐loop nonlinear system is guaranteed by means of a Lyapunov function that can clearly be seen to be the solution to the steady‐state form of the stochastic Hamilton–Jacobi–Bellman equation and, hence, guaranteeing both stochastic stability and optimality. In addition, we develop optimal feedback controllers for affine nonlinear systems using an inverse optimality framework tailored to the stochastic stabilization problem. These results are then used to provide extensions of the nonlinear feedback controllers obtained in the literature that minimize general polynomial and multilinear performance criteria. Copyright © 2017 John Wiley & Sons, Ltd.     

Mixed L2 and L∞ problems by weight selection in quadratic optimal control

In an attempt to achieve more realistic control objectives, the weighting matrices in the standard LQ1 problem are usually chosen by the designer in an ad hoc manner. This paper shows several optimal control design problems that minimize a quadratic function of the control vector subject to multiple inequality constraints on the output L ₂ norms, L _∞ norms, covariance matrix, and the maximum singular value of the output covariance matrix. The solutions of all four of these problems reduce to standard LQI control problems with different choices of weights. This paper shows how to construct these different weights. The practical significance of these results is that many robustness properties relate directly to these four entities. Hence the given control design algorithm delivers a specified degree of robustness to both parameter errors and disturbances. The results are presented in the deterministic terms of the linear quadratic impulse (LQI)for continuous and discrete systems problem rather than the stochastic LQG problem. The results are easily transferable to the LQG problem.     

An HMM approach for optimal investment of an insurer

In this paper, we introduce a Hidden Markov Model (HMM) for studying an optimal investment problem of an insurer when model uncertainty is present. More specifically, the financial price and insurance risk processes are modulated by a continuous‐time, finite‐state, hidden Markov chain. The states of the chain represent different modes of the model. The HMM approach is viewed as a ‘dynamic’ version of the Bayesian approach to model uncertainty. The optimal investment problem is formulated as a stochastic optimal control problem with partial observations. The innovations approach in the filtering theory is then used to transform the problem into one with complete observations. New robust filters of the chain and estimates of key parameters are derived. We discuss the optimal investment problem using the Hamilton–Jacobi–Bellman (HJB) dynamic programming approach and derive a closed‐form solution in the case of an exponential utility and zero interest rate. Copyright © 2011 John Wiley & Sons, Ltd.     

多阶段均值–方差资产负债管理的随机控制

资产负债管理研究如何合理分配资产以到达最小化风险同时确保期望剩余财富(财富减去负债)达到一定水平.本文在均值–方差投资组合理论的框架下研究两类资产负债管理模型, 包括带有跨期均值–方差投资目标和带有非破产约束的模型. 由于在动态规划意义下, 方差不具有可分性质, 传统的随机最优控制方法难以直接应用. 如采用处理动态均值–方差优化问题的嵌入法来解决以上问题会带来计算上的困难. 本文借鉴平均场控制的思想对以上两类问题加以研究. 本文假设了非常宽泛的市场模型: 所有的资产都是风险资产; 债务和风险资产之间存在相关性. 在此市场假设模型下, 本文给出了最优投资策略(控制率)的解析表达式和均值–方差有效前沿的表达形式. 本研究成果为投资者提供了新的投资策略, 可应用于更复杂的资产负债管理中.