Backward Stochastic Riccati Equations and Infinite Horizon L-Q Optimal Control with Infinite Dimensional State Space and Random Coefficients

We study the Riccati equation arising in a class of quadratic optimal control problems with infinite dimensional stochastic differential state equation and infinite horizon cost functional. We allow the coefficients, both in the state equation and in the cost, to be random. In such a context backward stochastic Riccati equations are backward stochastic differential equations in the whole positive real axis that involve quadratic non-linearities and take values in a non-Hilbertian space. We prove existence of a minimal non-negative solution and, under additional assumptions, its uniqueness. We show that such a solution allows to perform the synthesis of the optimal control and investigate its attractivity properties. Finally the case where the coefficients are stationary is addressed and an example concerning a controlled wave equation in random media is proposed.     

由Lévy过程驱动的一类特殊的高维的BSRDE解的存在唯一性

讨论由Brownian运动和Lévy过程共同驱动的线性随机系统的随机LQ问题,其中代价泛函是关于Lévy过程生成的σ-代数取条件期望.得到由Lévy过程驱动的新的多维的倒向随机Riccati方程,利用Bellman拟线性原理和单调收敛方法证明了此随机Riccati方程的解的存在性.     

A Deterministic Approach To Stochastic Optimal Control With Application To Anticipative Control

Using the decomposition of solution of SDE, we consider the stochastic optimal control problem with anticipative controls as a family of deterministic control problems parametrized by the paths of the driving Wiener process and of a newly introduced Lagrange multiplier stochastic process (nonanticipativity equality constraint). It is shown that the value function of these problems is the unique global solution of a robust equation (random partial differential equation) associated to a linear backward Hamilton-Jacobi-Bellman stochastic partial differential equation (HJB SPDE). This appears as limiting SPDE for a sequence of random HJB PDE's when linear interpolation approximation of the Wiener process is used. Our approach extends the Wong-Zakai type results [20] from SDE to the stochastic dynamic programming equation by showing how this arises as average of the limit of a sequence of deterministic dynamic programming equations. The stochastic characteristics method of Kunita [13] is used to represent the value function. By choosing the Lagrange multiplier equal to its nonanticipative constraint value the usual stochastic (nonanticipative) optimal control and optimal cost are recovered. This suggests a method for solving the anticipative control problems by almost sure deterministic optimal control. We obtain a PDE for the “cost of perfect information” the difference between the cost function of the nonanticipative control problem and the cost of the anticipative problem which satisfies a nonlinear backward HJB SPDE. Poisson bracket conditions are found ensuring this has a global solution. The cost of perfect information is shown to be zero when a Lagrangian submanifold is invariant for the stochastic characteristics. The LQG problem and a nonlinear anticipative control problem are considered as examples in this framework     

Optimal variational principle for backward stochastic control systems associated with Lévy processes

The paper is concerned with optimal control of backward stochastic differential equation (BSDE) driven by Teugel’s martingales and an independent multi-dimensional Brownian motion,where Teugel’s martin- gales are a family of pairwise strongly orthonormal martingales associated with Lévy processes (see e.g.,Nualart and Schoutens’ paper in 2000).We derive the necessary and sufficient conditions for the existence of the op- timal control by means of convex variation methods and duality techniques.As an application,the optimal control problem of linear backward stochastic differential equation with a quadratic cost criteria (or backward linear-quadratic problem,or BLQ problem for short) is discussed and characterized by a stochastic Hamilton system.     

A Finite Horizon Linear Quadratic Optimal Stochastic Control Problem Driven by Both Brownian Motion and L\'{e}vy Processes

We study the linear quadratic optimal stochastic control problem which is jointly driven by Brownian motion and L\'{e}vy processes. We prove that the new affine stochastic differential adjoint equation exists an inverse process by applying the profound section theorem. Applying for the Bellman's principle of quasilinearization and a monotone iterative convergence method, we prove the existence and uniqueness of the solution of the backward Riccati differential equation. Finally, we prove that the optimal feedback control exists, and the value function is composed of the initial value of the solution of the related backward Riccati differential equation and the related adjoint equation.     

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??We study the linear quadratic optimal stochastic control problem which is jointly driven by Brownian motion and L\'{e}vy processes. We prove that the new affine stochastic differential adjoint equation exists an inverse process by applying the profound section theorem. Applying for the Bellman's principle of quasilinearization and a monotone iterative convergence method, we prove the existence and uniqueness of the solution of the backward Riccati differential equation. Finally, we prove that the optimal feedback control exists, and the value function is composed of the initial value of the solution of the related backward Riccati differential equation and the related adjoint equation.     

随机递归最优控制和混合最优控制问题

对随机递归最优控制问题即代价函数由特定倒向随机微分方程解来描述和递归混合最优控制问题即控制者还需 决定最优停止时刻, 得到了最优控制的存在性结果. 在一类等价概率测度集中,还给出了递归最优值函数的最小和最大数学期望.     

Optimal control of backward stochastic heat equation with Neumann boundary control and noise

This paper considers a stochastic control problem in which the dynamic system is a controlled backward stochastic heat equation with Neumann boundary control and boundary noise and the state must coincide with a given random vector at terminal time. Through defining a proper form of the mild solution for the state equation, the existence and uniqueness of the mild solution is given. As a main result, a global maximum principle for our control problem is presented. The main result is also applied to a backward linear-quadratic control problem in which an optimal control is obtained explicitly as a feedback of the solution to a forward–backward stochastic partial differential equation.     

General Linear Quadratic Optimal Stochastic Control Problem Driven by a Brownian Motion and a Poisson Random Martingale Measure with Random Coefficients

In this article, we consider a linear-quadratic optimal control problem (LQ problem) for a controlled linear stochastic differential equation driven by a multidimensional Browinan motion and a Poisson random martingale measure in the general case, where the coefficients are allowed to be predictable processes or random matrices. By the duality technique, the dual characterization of the optimal control is derived by the optimality system (so-called stochastic Hamilton system), which turns out to be a linear fully coupled forward-backward stochastic differential equation with jumps. Using a decoupling technique, the connection between the stochastic Hamilton system and the associated Riccati equation is established. As a result, the state feedback representation is obtained for the optimal control. As the coefficients for the LQ problem are random, here, the associated Riccati equation is a highly nonlinear backward stochastic differential equation (BSDE) with jumps, where the generator depends on the unknown variables K, L, and H in a quadratic way (see (5.9) herein). For the case where the generator is bounded and is linearly dependent on the unknown martingale terms L and H, the existence and uniqueness of the solution for the associated Riccati equation are established by Bellman's principle of quasi-linearization.     

A variational formula for controlled backward stochastic partial differential equations and some application

An optimal control problem for a controlled backward stochastic partial differential equation in the abstract evolution form with a Bolza type performance functional is considered. The control domain is not assumed to be convex, and all coefficients of the system are allowed to be random. A variational formula for the functional in a given control process direction is derived, by the Hamiltonian and associated adjoint system. As an application, a global stochastic maximum principle of Pontraygins type for the optimal controls is established.     

A BSDE approach to Nash equilibrium payoffs for stochastic differential games with nonlinear cost functionals

In this paper, we study Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games via the theory of backward stochastic differential equations. We obtain an existence theorem and a characterization theorem of Nash equilibrium payoffs for two-player nonzero-sum stochastic differential games with nonlinear cost functionals defined with the help of doubly controlled backward stochastic differential equations. Our results extend former ones by Buckdahn et al. (2004) [3] and are based on a backward stochastic differential equation approach.     

Kalman-Bucy filtering equations of forward and backward stochastic systems and applications to recursive optimal control problems

This paper is concerned with Kalman-Bucy filtering problems of a forward and backward stochastic system which is a Hamiltonian system arising from a stochastic optimal control problem. There are two main contributions worthy pointing out. One is that we obtain the Kalman-Bucy filtering equation of a forward and backward stochastic system and study a kind of stability of the aforementioned filtering equation. The other is that we develop a backward separation technique, which is different to Wonham's separation theorem, to study a partially observed recursive optimal control problem. This new technique can also cover some more general situation such as a partially observed linear quadratic non-zero sum differential game problem is solved by it. We also give a simple formula to estimate the information value which is the difference of the optimal cost functionals between the partial and the full observable information cases.     

Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability

In this paper we study the existence of the optimal (minimizing) control for a tracking problem, as well as a quadratic cost problem subject to linear stochastic evolution equations with unbounded coefficients in the drift. The backward differential Riccati equation (BDRE) associated with these problems (see [2], for finite dimensional stochastic equations or [21], for infinite dimensional equations with bounded coefficients) is in general different from the conventional BDRE (see [10], [18]). Under stabilizability and uniform observability conditions and assuming that the control weight-costs are uniformly positive, we establish that BDRE has a unique, uniformly positive, bounded on ℝ ₊ and stabilizing solution. Using this result we find the optimal control and the optimal cost. It is known [18] that uniform observability does not imply detectability and consequently our results are different from those obtained under detectability conditions (see [10]).      

Linear-quadratic optimal control under non-Markovian switching

We study a finite-dimensional continuous-time optimal control problem on finite horizon for a controlled diffusion driven by Brownian motion, in the linear-quadratic case. We admit stochastic coefficients, possibly depending on an underlying independent marked point process, so that our model is general enough to include controlled switching systems where the switching mechanism is not required to be Markovian. The problem is solved by means of a Riccati equation, which turned out to be a backward stochastic differential equation driven by the Brownian motion and by the random measure associated with the marked point process.     

Stochastic Optimal Control for the Stochastic Heat Equation with Exponentially Growing Coefficients and with Control and Noise on a Subdomain

Abstract

We consider stochastic optimal control problems in Banach spaces, related to nonlinear controlled equations with dissipative non linearities: on the nonlinear term we do not impose any growth condition. The problems are treated via the backward stochastic differential equations approach, that allows also to solve in mild sense Hamilton Jacobi Bellman equations in Banach spaces. We apply the results to controlled stochastic heat equation, in space dimension 1, with control and noise acting on a subdomain.     

Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise

We consider the controlled stochastic Navier–Stokes equations in a bounded multidimensional domain, where the noise term allows jumps. In order to prove existence and uniqueness of an optimal control w.r.t. a given control problem, we first need to show the existence and uniqueness of a local mild solution of the considered controlled stochastic Navier–Stokes equations. We then discuss the control problem, where the related cost functional includes stopping times dependent on controls. Based on the continuity of the cost functional, we can apply existence and uniqueness results provided in [4], which enables us to show that a unique optimal control exists.     

L\'evy过程驱动的正倒向随机系统的随机最大值原理

研究了由Teugels鞅和与之独立的多维Brown运动共同驱动的正倒向随机控制系统的最优控制问题. 这里Teugels鞅是一列与L\'{e}vy 过程相关的两两强正交的正态鞅 (见Nualart, Schoutens 在2000年的结果). 在允许控制值域为一非空凸闭集假设下, 采用凸变分法和对偶技术获得了最优控制存在所满足的充分和必要条件. 作为应用, 系统研究了线性正倒向随机系统的二次最优控制问题(简记为FBLQ问题), 通过相应的随机哈密顿系统对最优控制 进行了对偶刻画. 这里的随机哈密顿系统是由Teugels鞅和多维Brown运动共同驱动的线性正倒向随机微分方程, 其由状态方程、伴随方程和最优控制的对偶表示共同来构成.     

Forward and Backward Mean-Field Stochastic Partial Differential Equation and Optimal Control

This paper is mainly concerned with the solutions to both forward and backward mean-field stochastic partial differential equation and the corresponding optimal control problem for mean-field stochastic partial differential equation. The authors first prove the continuous dependence theorems of forward and backward mean-field stochastic partial differential equations and show the existence and uniqueness of solutions to them. Then they establish necessary and sufficient optimality conditions of the control problem in the form of Pontryagin''s maximum principles. To illustrate the theoretical results, the authors apply stochastic maximum principles to study the infinite-dimensional linear-quadratic control problem of mean-field type. Further, an application to a Cauchy problem for a controlled stochastic linear PDE of mean-field type is studied.     

分数布朗运动驱动下z一致连续的BSDE解的存在性与唯一性

本文研究一类由分数布朗运动驱动的一维倒向随机微分方程解的存在性与唯一性问题,在假设其生成元满足关于y Lipschitz连续,但关于z一致连续的条件下,通过应用分数布朗运动的Tanaka公式以及拟条件期望在一定条件下满足的单调性质,得到倒向随机微分方程的解的一个不等式估计,应用Gronwall不等式得到了一个关于这类方程的解的存在性与唯一性结果,推广了一些经典结果以及生成元满足一致Lipschitz条件下的由分数布朗运动驱动的倒向随机微分方程解的结果.     

Ergodic BSDEs under weak dissipative assumptions

In this paper we study ergodic backward stochastic differential equations (EBSDEs) dropping the strong dissipativity assumption needed in Fuhrman et al. (2009) [12]. In other words we do not need to require the uniform exponential decay of the difference of two solutions of the underlying forward equation, which, on the contrary, is assumed to be non-degenerate.We show the existence of solutions by the use of coupling estimates for a non-degenerate forward stochastic differential equation with bounded measurable nonlinearity. Moreover we prove the uniqueness of “Markovian” solutions by exploiting the recurrence of the same class of forward equations.Applications are then given for the optimal ergodic control of stochastic partial differential equations and to the associated ergodic Hamilton-Jacobi-Bellman equations.