条件平均场随机微分方程的最优控制问题

作者研究了一个条件平均场随机微分方程的最优控制问题.这种方程和某些部分信息下的随机最优控制问题有关,并且可以看做是平均场随机微分方程的推广.作者以庞特里雅金最大值原理的形式给出最优控制满足的必要和充分条件.此外,文中给出一个线性二次最优控制问题来说明理论结果的应用.     

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.     

Forward-backward Stochastic Differential Equations and Backward Linear Quadratic Stochastic Optimal Control Problem

In this paper, we use the solutions of forward-backward stochastic differential equations to get the optimal control for backward stochastic linear quadratic optimal control problem. And we also give the linear feedback regulator for the optimal control problem by using the solutions of a group of Riccati equations.     

一类无穷区间的倒向随机微分方程及其应用

本文讨论了一类基于无穷区间的倒向随机微分方程解的存在唯一性及其性质. 由方程解定义一类非线性g-期望, 并讨论其在经济金融中的应用.     

由L\'{e}vy过程驱动的仿射方程关联的无限时区的最优二次控制

讨论线性二次最优控制问题, 其随机系统是由 L\'{e}vy 过程驱动的具有随机系数而且还具有仿射项的线性随机微分方程. 伴随方程具有无界系数, 其可解性不是显然的. 利用 $\mathscr{B}\mathscr{M}\mathscr{O}$ 鞅理论, 证明伴随方程在有限 时区解的存在唯一性. 在稳定性条件下, 无限时区的倒向随机 Riccati 微分方程和伴随倒向随机方程的解的存在性是通过对应有限 时区的方程的解来逼近的. 利用这些解能够合成最优控制.     

非Lipschitz条件下的倒向重随机微分方程

该文研究了非Lipschitz条件下的倒向重随机微分方程, 给出了此类方程解的存在唯一性 定理, 推广Pardoux和Peng 1994年的结论; 同时也得到了此类方程在非Lipschitz条件下的比较定理, 推广了Shi,Gu和Liu 2005年的结果. 从而推广倒向重随机微分方程在随机控制和随机偏微分方程在 粘性解方面的应用.     

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.     

平均场倒向重随机微分方程及其应用

研究了平均场倒向随重机微分方程,得到了平均场倒向重随机微分方程解的存在唯一性.基于平均场倒向重随机微分方程的解,给出了一类非局部随机偏微分方程解的概率解释.讨论了平均场倒向重随机系统的最优控制问题,建立了庞特利亚金型的最大值原理.最后讨论了一个平均场倒向重随机线性二次最优控制问题,展示了上述最大值原理的应用.     

随机系数下线性二次控制问题的最优性条件及其应用*

本文旨在研究随机系数下随机微分方程的线性二次最优控制问题.本文从闭环最优控制/策略存在的必要性条件的角度开展研究. 若闭环最优控制/策略存在, 得到其显示反馈表示、带伪逆运算的倒向随机Riccati方程的适定性及不同系数间满足的一些本质性条件. 此处结论本质地推广和改进了文[Ait Rami M, Moore J, Zhou X. Indefinite stochastic linear quadratic control and generalized differential Riccati equation [J]. {\it SIAM J Control Optim,} 2001, 40:1296--1311;Sun J, Yong J. Linear quadratic stochastic differential games: open-loop and closed-loop saddle points [J]. {\it SIAM J Control Optim,} 2014, 52:4082--4121;L\"{u} Q, Wang T, Zhang X. Characterization of optimal feedback for stochastic linear quadratic control problems,Probab Uncertain Quant Risk, 2017, 2017, 2:11, DOI 10.1186/s41546-017-0022-7]的相应结论.此外, 本文得到了一个关于倒向随机Riccati方程和二阶伴随方程两类方程适应解之间的微妙关系. 注意到,这一结论在现有文献中首次出现. 最后, 本文讨论了在均值方差对冲问题中的应用.     

带奇异系数的有限维与无穷维随机微分方程

本文对具非Lipschitz系数的随机微分方程给出解的存在唯一性与非爆炸性的新判别条件,少许改进了文\cite{4}的有关结果. 通过控制交互作用, 该结果还被推广到无穷维情形.     

无穷水平的随机微分效用

本文研究了由Ｄｕｆｆｉｅ－Ｅｐｓｔｅｉｎ提出的无穷水平的随机微分效用理论,建立了无穷水平的随机微分效用和无穷限倒抽随机微分方程的等价关系。在非－Ｌｉｐｓｃｈｉｔｚ条件下,讨论了无穷水平的随机微分效用的存在唯一性和效用函数的一系列效用。     

An Infinite Horizon Linear Quadratic Problem with Unbounded Controls in Hilbert Space

An infinite horizon linear quadratic optimal control problem for analytic semigroup with unbounded control in Hilbert space is considered. The state weight operator is allowed to be indefinite while the control weight operator is coercive. Under the exponential stabilization condition, it is proved that any optimal control and its optimal trajectory are continuous. The positive real lemma as a necessary and sufficient condition for the unique solvability of this problem is established. The closed-loop synthesis of optimal control is given via the solution to the algebraic Riccati equation. This work is partially supported by the National Key Project of China, the National Nature Science Foundation of China No. 19901030, NSF of the Chinese State Education Ministry and Lab. of Math. for Nonlinear Sciences at Fudan University     

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.     

随机线性二次最优控制:从离散到连续时间模型

在一般情形下,分析了离散时间LQ问题与连续时间情形两者之间的自然联系.首先回顾了连续时间和离散时间随机LQ问题及对应Riccati微分/差分方程的相关结论.接下来在假设Riccati微分方程有解的前提下,证明了离散化步长足够小时,Riccati差分方程有解.然后针对连续和离散时间模型,采用配对问题最优控制的反馈形式,分别构造了一个辅助反馈控制,并证明该控制可驱使对应模型的性能指标逼近于配对问题的值函数,以此得到了关于两个模型之间联系的初步结论.最后藉由前述结论以及控制问题的特性,揭晓了连续时间和离散时间模型之间的自然联系,并给出了Riccati差分方程和微分方程的解之间的误差估计.由此联系,可构造相应离散系统和LQ问题,以适当的阶估计连续时间LQ问题的解,抑或为离散时间模型构造一个近似最优控制.无论哪种思路,都旨在降低直接求解原问题的难度和复杂性.     

The Filtering Equations of Forward-Backward Stochastic Systems with Random Jumps and Applications to Partial Information Stochastic Optimal Control

In this article, we consider a filtering problem for forward-backward stochastic systems that are driven by Brownian motions and Poisson processes. This kind of filtering problem arises from the study of partially observable stochastic linear-quadratic control problems. Combining forward-backward stochastic differential equation theory with certain classical filtering techniques, the desired filtering equation is established. To illustrate the filtering theory, the theoretical result is applied to solve a partially observable linear-quadratic control problem, where an explicit observable optimal control is determined by the optimal filtering estimation.     

Backward Doubly Stochastic Differential Equations with Jumps and Stochastic Partial Differential-Integral Equations

Backward doubly stochastic differential equations driven by Brownian motions and Poisson process(BDSDEP) with non-Lipschitz coeffcients on random time interval are studied.The probabilistic interpretation for the solutions to a class of quasilinear stochastic partial differential-integral equations(SPDIEs) is treated with BDSDEP.Under non-Lipschitz conditions,the existence and uniqueness results for measurable solutions to BDSDEP are established via the smoothing technique.Then,the continuous dependence for solutions to BDSDEP is derived.Finally,the probabilistic interpretation for the solutions to a class of quasilinear SPDIEs is given.     

Two-Step Scheme for Backward Stochastic Differential Equations

In this paper, a stochastic linear two-step scheme has been presented to approximate backward stochastic differential equations (BSDEs). A necessary and sufficient condition is given to judge the $\mathbb{L}_2$-stability of our numerical schemes. This stochastic linear two-step method possesses a family of $3$-order convergence schemes in the sense of strong stability. The coefficients in the numerical methods are inferred based on the constraints of strong stability and $n$-order accuracy ($n\in\mathbb{N}^+$). Numerical experiments illustrate that the scheme is an efficient probabilistic numerical method.     

A Stochastic Linear Quadratic Optimal Control Problem with Generalized Expectation

Abstract

In this article, we initiate a study on optimal control problem for linear stochastic differential equations with quadratic cost functionals under generalized expectation via backward stochastic differential equations.     

Semilinear Kolmogorov Equations and Applications to Stochastic Optimal Control

Semilinear parabolic differential equations are solved in a mild sense in an infinite-dimensional Hilbert space. Applications to stochastic optimal control problems are studied by solving the associated Hamilton–Jacobi–Bellman equation. These results are applied to some controlled stochastic partial differential equations.     

求解无限维线性二次最优控制问题的一种新的途径

本文借助Ｃ_０半群的Ｙｏｓｉｄａ近似构造无限维线性二次最优控制问题的相应近似，证明了后者的最优控制、Ｒｉｃｃａｔｉ方程之解（从而反馈算子）和最优状态函数均一致强收敛，极限即为原问题的解．