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．     

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.     

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.     

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.     

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.     

Linear−quadratic optimal control and nonzero‐sum differential game of forward−backward stochastic system

An existence and uniqueness result for one kind of forward–backward stochastic differential equations with double dimensions was obtained under some monotonicity conditions. Then this result was applied to the linear‐quadratic stochastic optimal control and nonzero‐sum differential game of forward–backward stochastic system. The explicit forms of the optimal control and the Nash equilibrium point are obtained respectively. We note that our method is effective in studying the uniqueness of Nash equilibrium point. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

On the Controllability of Nonlocal Second‐Order Impulsive Neutral Stochastic Integro‐Differential Equations with Infinite Delay

In this paper, we investigate the controllability for a class of nonlocal second‐order impulsive neutral stochastic integro‐differential equations with infinite delay in Hilbert spaces. More precisely, a set of sufficient conditions for the controllability results of nonlocal second‐order impulsive neutral stochastic integro‐differential equations with infinite delay are derived by means of the Banach fixed point theorem combined with theories of a strongly continuous cosine family of bounded linear operators. As an application, an example is provided to illustrate the obtained theory.     

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.     

Existence of Optimal Mild Solutions and Controllability of Fractional Impulsive Stochastic Partial Integro‐Differential Equations with Infinite Delay

In this paper, we investigate a new class of fractional impulsive stochastic partial integro‐differential equations with infinite delay in Hilbert spaces. By using the stochastic analysis theory, fractional calculus, analytic α‐resolvent operator and the fixed point technique combined with fractional powers of closed operators, we firstly give the existence of of mild solutions and optimal mild solutions for the these equations. Next, the controllability of the controlled fractional impulsive stochastic partial integro‐differential systems with not instantaneous impulses is presented. Finally, examples are also given to illustrate our results.     

Multi-valued stochastic differential equations driven by G-Brownian motion and related stochastic control problems

In this paper, we prove the existence and uniqueness of a solution for a class of multi-valued stochastic differential equations driven by G-Brownian motion (MSDEG) by means of the Yosida approximation method. Moreover, we set up an optimality principle of stochastic control problem and prove the value function of the control problem is the unique viscosity solution of a class of nonlinear partial differential variational inequalities.     

Stochastic maximum principle for delayed doubly stochastic control systems and their applications

ABSTRACT

In this paper, we investigate the optimal control problems for delayed doubly stochastic control systems. We first discuss the existence and uniqueness of the delayed doubly stochastic differential equation by martingale representation theorem and contraction mapping principle. As a necessary condition of the optimal control, we deduce a stochastic maximum principle under some assumption. At the same time, a sufficient condition of optimality is obtained by using the duality method. At the end of the paper, we apply our stochastic maximum principle to a class of linear quadratic optimal control problem and obtain the explicit expression of the optimal control.     

Stability of Impulsive Stochastic Neutral Partial Differential Equations With Infinite Delays

In this paper, we study the existence and asymptotic stability in the pth moment of the mild solutions to impulsive stochastic neutral partial differential equations with infinite delays. Sufficient conditions ensuring the stability of the impulsive stochastic system are established. The results are obtained via the Banach fixed point theorem.     

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.     

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.     

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.     

Global asymptotic stabilisation in probability of nonlinear stochastic systems via passivity

The purpose of this paper is to develop a systematic method for global asymptotic stabilisation in probability of nonlinear control stochastic systems with stable in probability unforced dynamics. The method is based on the theory of passivity for nonaffine stochastic differential systems combined with the technique of Lyapunov asymptotic stability in probability for stochastic differential equations. In particular, we prove that a nonlinear stochastic differential system whose unforced dynamics are Lyapunov stable in probability is globally asymptotically stabilisable in probability provided some rank conditions involving the affine part of the system coefficients are satisfied. In this framework, we show that a stabilising smooth state feedback law can be designed explicitly. A dynamic output feedback compensator for a class of nonaffine stochastic systems is constructed as an application of our analysis.     

Existence and Controllability Results for a New Class of Impulsive Stochastic Partial Integro‐Differential Inclusions with State‐Dependent Delay

In this paper, we study a new class of impulsive stochastic partial integro‐differential inclusions with state‐dependent delay in separable Hilbert spaces. Firstly, by using stochastic analysis theory, analytic resolvent operators, fractional powers of closed operators and suitable fixed point theorems, we prove the existence of mild and extremal mild solutions for these systems in the α‐norm. Secondly, we establish the controllability of the controlled stochastic partial integro‐differential inclusions with not instantaneous impulses. The results are obtained under the mixed Lipschitz and Carathéodory conditions. Finally, an example is provided to show the application of our results.     

pTH Moment Exponential Stability of Stochastic Partial Differential Equations with Poisson Jumps

In this paper, we study a class of stochastic partial differential equations with Poisson jumps, which is more realistic for establishing mathematical models since it has been widely applied in many fields. Under a reasonable condition, we not only establish the existence and uniqueness of the mild solution for the investigated system but also prove that it is pth moment exponentially stable by using the fixed point theory. Then, based on the well‐known Borel‐Cantelli lemma, further, we prove that the mild solution is almost surely pth moment exponentially stable. Our results improve and generalize those given in the previous literature, in particular, the Lyapunov direct method and successive approximation method. Finally, we give an example to illustrate the effectiveness of the obtained results.     

Verification Theorem Of Stochastic Optimal Control With Mixed Delay And Applications To Finance

This paper focuses on a general model of a controlled stochastic differential equation with mixed delay in the state variable. Based on the Itô formula, stochastic analysis, convex analysis, and inequality technique, we obtain a semi‐coupled forward‐backward stochastic differential equation with mixed delay and mixed initial‐terminal conditions and prove that such forward‐backward system admits a unique adapted solution. The verification theorem for an optimal control of a system with mixed delay is established. The obtained results generalize and improve some recent results, and they are more easily verified and applied in practice. As an application, we conclude with finding explicitly the optimal consumption rate from the wealth process of a person given by a stochastic differential equation with mixed delay which fit into our general model.     

Recursive Stochastic H2/H∞ Control Problem for Delay Systems Involving Continuous and Impulse Controls

In this paper, we discuss the recursive stochastic H₂/H_∞ control problem of delay systems with random coefficients involving both continuous and impulse controls. By virtue of a new type of forward backward stochastic differential equations, a necessary and sufficient condition for the existence of a unique solution to the control problem under consideration is derived. The existence and uniqueness of the forward backward stochastic differential equations are also be proved.