An infinite horizon stochastic maximum principle for discounted control problem with Lipschitz coefficients

In the present work, a stochastic maximum principle for discounted control of a certain class of degenerate diffusion processes with global Lipschitz coefficient is investigated. The value function is given by a discounted performance functional, leading to a stochastic maximum principle of semi-couple forward–backward stochastic differential equation with non-smooth coefficients. The proof is based on the approximation of the Lipschitz coefficients by smooth ones and the approximation of the infinite horizon adjoint process.     

Mean-field type forward-backward doubly stochastic differential equations and related stochastic differential games

We study a kind of partial information non-zero sum differential games of mean-field backward doubly stochastic differential equations, in which the coefficient contains not only the state process but also its marginal distribution, and the cost functional is also of mean-field type. It is required that the control is adapted to a sub-filtration of the filtration generated by the underlying Brownian motions. We establish a necessary condition in the form of maximum principle and a verification theorem, which is a sufficient condition for Nash equilibrium point. We use the theoretical results to deal with a partial information linear-quadratic (LQ) game, and obtain the unique Nash equilibrium point for our LQ game problem by virtue of the unique solvability of mean-field forward-backward doubly stochastic differential equation.     

Impulsive optimal control with finite or infinite time horizon

We consider a dynamical system subjected to feedback optimal control in such a way that the evolution of the state exhibits both sudden jumps and continuous changes. Previously obtained necessary conditions (Ref. 1) for such impulsive optimal feedback controls are generalized to admit the case of infinite time horizon; this generalization permits application to a wider class of problems. The results are illustrated by application to a version of the innkeeper's problem.Dedicated to G. Leitmann     

A maximum principle for optimal control problem of fully coupled forward-backward stochastic systems with partial information

The paper is concerned with a stochastic optimal control problem in which the controlled system is described by a fully coupled nonlinear forward-backward stochastic differential equation driven by a Brownian motion. It is required that all admissible control processes are adapted to a given subfiltration of the filtration generated by the underlying Brownian motion. For this type of partial information control, one sufficient (a verification theorem) and one necessary conditions of optimality are proved. The control domain need to be convex and the forward diffusion coefficient of the system can contain the control variable. This work was partially supported by Basic Research Program of China (Grant No. 2007CB814904), National Natural Science Foundation of China (Grant No. 10325101) and Natural Science Foundation of Zhejiang Province (Grant No. Y605478, Y606667)     

A maximum principle for fully coupled stochastic control systems of mean-field type

The present paper considers an optimal control problem for fully coupled forward–backward stochastic differential equations (FBSDEs) of mean-field type in the case of controlled diffusion coefficient. Moreover, the control domain is not assumed to be convex. By virtue of a reduction method, we establish the necessary optimality conditions of Pontryagin's type. As an application, a linear–quadratic stochastic control problem is studied.     

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.     

倒向随机微分方程与巴黎期权的非线性定价

巴黎期权是一种复杂的奇异期权. 本文基于倒向随机微分方程, 定义了巴黎期权的非线性价格过程, 分析其性质, 并且给出巴黎期权非线性定价的偏微分方程表达式. 在金融市场收益率不确定的情形以及存贷利率不同的情形下分别对连续巴黎期权进行定价和具体的数值分析, 结论显示巴黎期权的非线性定价机制更具合理性.     

一类状态受限的随机延迟最优控制问题的最大值原理

本文考虑一类状态受限的随机延迟最优控制问题,其中控制域为凸集且扩散项系数中含有控制变量.控制域可以是无界集合.用最大值原理方法建立了最优控制满足的必要条件.也给出了充分最优性条件,从而有助于找到最优控制.     

Large deviation for mean-field stochastic differential equations with subdifferential operator

In this article, we prove the existence and uniqueness of a solution for a class of mean-field stochastic differential equations with subdifferential operator (i.e., mean-field MSDEs) by means of the Moreau–Yosida type penalization method. Moreover, we prove a large deviation principle of its path solution via the weak convergence method.     

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.     

Approximating infinite horizon stochastic optimal control in discrete time with constraints

Traditional approaches to solving stochastic optimal control problems involve dynamic programming, and solving certain optimality equations. When recast as stochastic programming problems, structural aspects such as convexity are retained, and numerical solution procedures based on decomposition and duality may be exploited. This paper explores a class of stationary, infinite-horizon stochastic optimization problems with discounted cost criterion. Constraints on both states and controls are permitted, and modeled in the objective function by allowing it to take infinite values. Approximating techniques are developed using variational analysis, and intuitive lower bounds are obtained via averaging the future. These bounds could be used in a finite-time horizon stochastic programming setting to find solutions numerically. Research supported in part by a grant of the National Science Foundation. AMS Classification 46N10, 49N15, 65K10, 90C15, 90C46     

Forward-backward stochastic differential equation games with delay and noisy memory

Abstract

The goal of this paper is to study a stochastic game connected to a system of forward-backward stochastic differential equations (FBSDEs) involving delay and noisy memory. We derive sufficient and necessary maximum principles for a set of controls for the players to be a Nash equilibrium in the game. Furthermore, we study a corresponding FBSDE involving Malliavin derivatives. This kind of equation has not been studied before. The maximum principles give conditions for determining the Nash equilibrium of the game. We use this to derive a closed form Nash equilibrium for an economic model where the players maximize their consumption with respect to recursive utility.     

Stationary solutions of SPDEs and infinite horizon BDSDEs

In this paper we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between valued solutions of backward doubly stochastic differential equations (BDSDEs) on infinite horizon and the stationary solutions of the SPDEs. Moreover, we prove the existence and uniqueness of the solutions of BDSDEs on both finite and infinite horizons, so obtain the solutions of initial value problems and the stationary solutions (independent of any initial value) of SPDEs. The connection of the weak solutions of SPDEs and BDSDEs has independent interests in the areas of both SPDEs and BSDEs.     

Limit theorem and uniqueness theorem of backward stochastic differential equations

This paper establishes a limit theorem for solutions of backward stochastic differential equations (BSDEs). By this limit theorem, this paper proves that, under the standard assumption g(t,y,0) = 0, the generator g of a BSDE can be uniquely determined by the corresponding g-expectationεg;this paper also proves that if a filtration consistent expectation S can be represented as a g-expectationεg, then the corresponding generator g must be unique.     

Duality theorem for the stochastic optimal control problem

We prove a duality theorem for the stochastic optimal control problem with a convex cost function and show that the minimizer satisfies a class of forward–backward stochastic differential equations. As an application, we give an approach, from the duality theorem, to h$h$-path processes for diffusion processes.     

On optimal solutions of general continuous-singular stochastic control problem of McKean-Vlasov type

In this paper, we establish general necessary optimality conditions for stochastic continuous-singular control of McKean-Vlasov type equations. The coefficients of the state equation depend on the state of the solution process as well as of its probability law and the control variable. The coefficients of the system are nonlinear and depend explicitly on the absolutely continuous component of the control. The control domain under consideration is not assumed to be convex. The proof of our main result is based on the first- and second-order derivatives, with respect to measure in Wasserstein space of probability measures, and by using variational method.     

Ergodicity for neutral type SDEs with infinite length of memory

In this paper, the weak Harris theorem developed in [18] is illustrated by using a straightforward Wasserstein coupling, which implies the exponential ergodicity of the functional solutions to a range of neutral type SDEs with infinite length of memory. A concrete example is presented to illustrate the main result.     

带Poisson 跳跃的正倒向随机延迟系统递归最优控制问题的最大值原理

本文研究了带Poisson 跳跃的正倒向随机延迟系统的递归最优控制问题. 利用经典的针状变分方法、对偶技术和带Poisson 跳跃的超前倒向随机微分方程的相关结果, 证明了最优控制的最大值原理, 包括了最优控制满足的必要条件和充分条件.