Max-Plus Stochastic Processes

This paper is concerned with processes which are max-plus counterparts of Markov diffusion processes governed by Ito sense stochastic differential equations. Concepts of max-plus martingale and max-plus stochastic differential equation are introduced. The max-plus counterparts of backward and forward PDEs for Markov diffusions turn out to be first-order PDEs of Hamilton–Jacobi–Bellman type. Max-plus additive integrals and a max-plus additive dynamic programming principle are considered. This leads to variational inequalities of Hamilton–Jacobi–Bellman type.     

Max-Plus Algebra and Mathematical Fear in Dynamic Optimization

Max-plus algebra, cost measures, and mathematical fear have proved useful tools in dynamic optimization. Indeed, the first two have even become a central tool in some fields of investigation such as discrete event systems. We first recall the fundamentals of max-plus algebra with simple examples of max-plus linear models, and simple consequences of that remark. We then introduce cost measures, the natural equivalent of probability measures in the max-plus algebra, and their fundamental properties, including the definition of the mathematical fear (the equivalent of the mathematical expectation), induced measures and conditioning. Finally, we concentrate on those aspects that are put in use in dynamical optimization and state a separation theorem which was first derived using these tools.     

Max-Plus Objects to Study the Complexity of Graphs

Given an undirected graph G, we define a new object H _G , called the mp-chart of G, in the max-plus algebra. We use it, together with the max-plus permanent, to describe the complexity of graphs. We show how to compute the mean and the variance of H _G in terms of the adjacency matrix of G and we give a central limit theorem for H _G . Finally, we show that the mp-chart is easily tractable also for the complement graph.     

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.     

Computing an Eigenvector of a Monge Matrix in Max-Plus Algebra

The problem of finding one eigenvector of a given Monge matrix A in a max-plus algebra is considered. For a general matrix, the problem can be solved in O(n ³) time by computing one column of the corresponding metric matrix Δ(A _λ), where λ is the eigenvalue of A. An algorithm is presented, which computes an eigenvector of a Monge matrix in O(n ²) time.     

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.     

库存控制的随机优化(英文)

本文将随机扰动引入企业库存控制系统,并重点讨论了扰动引入后,库存投资与市场需求的相互作用,及对未来折现利润的相关影响.文章建立了关于库存控制管理的最优跨时决策模型.不同于其它文献中将市场需求及库存投资看作具有具体分布的随机变量,文章将两者设置为随机过程并分别服从不同的It型SDE.文章得出了三个命题及一些有意义的结果.     

极大加可约矩阵的谱与周期时间向量的关系

提出了极大加代数上可约矩阵特征值的缺失值及冗余值的概念,得到了相应的定理;对特征值与周期时间向量分量之间的关系作了深入的研究.     

Stochastic Optimal Control and Linear Programming Approach

We study a classical stochastic optimal control problem with constraints and discounted payoff in an infinite horizon setting. The main result of the present paper lies in the fact that this optimal control problem is shown to have the same value as a linear optimization problem stated on some appropriate space of probability measures. This enables one to derive a dual formulation that appears to be strongly connected to the notion of (viscosity sub) solution to a suitable Hamilton-Jacobi-Bellman equation. We also discuss relation with long-time average problems.     

Stochastic Control for Mean-Field Stochastic Partial Differential Equations with Jumps

We study optimal control for mean-field stochastic partial differential equations (stochastic evolution equations) driven by a Brownian motion and an independent Poisson random measure, in case of partial information control. One important novelty of our problem is represented by the introduction of general mean-field operators, acting on both the controlled state process and the control process. We first formulate a sufficient and a necessary maximum principle for this type of control. We then prove the existence and uniqueness of the solution of such general forward and backward mean-field stochastic partial differential equations. We apply our results to find the explicit optimal control for an optimal harvesting problem.     

Combined Optimal Stopping and Singular Stochastic Control

In this article, a simple of combined singular stochastic control and optimal stopping in the jump-diffusion model is formulated and solved. We give sufficient conditions for the existence of an optimal strategy which has the same form as in continuous case given by Davis and Zervos [3 Davis , M.H.A. , and Zervos , M. 1994. A problem of singular stochastic control with discretionary stopping. Annals of Applied Probility 4(1):226–240. [Google Scholar]] and also Karatzas et al. [5 Karatzas , I. , Ocone , D. , Wang , H. , and Zervos , M. 2000 . Finite-fuel singular control with discretionary stopping . Stochastics and Stochastics Reports 71 : 1 – 50 .[Taylor & Francis Online] , [Google Scholar]]. This result is applied to solve explicitly an example of such problem.     

极大-加混合线性不等式系统的可解性及其应用

研究极大-加混合线性不等式系统的可解性.基于极大-加线性方程系统可解的特征以及极大-加混合线性不等式系统的最大解,给出极大-加混合线性不等式系统可解的一个充分必要条件,还给出极大-加混合线性不等式系统在部分变量非负的约束条件下可解的一个充分必要条件.同时,例举一个制造系统加工工件时序规划的应用例子.     

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.     

Dynamics Aggregation in Stochastic Control Problems

For stochastic control problems with mixed state-control constraints, we develop a dynamics aggregation method which replaces the stochastic differential or difference equation with a simpler constraint. Solutions of such simplified problems are used to construct a sequence of approximations to the original problem. Convergence properties of the method for both discrete-time and continuous-time models are analyzed in detail.     

New Approach to Stochastic Optimal Control

This paper provides new insights into the solution of optimal stochastic control problems by means of a system of partial differential equations, which characterize directly the optimal control. This new system is obtained by the application of the stochastic maximum principle at every initial condition, assuming that the optimal controls are smooth enough. The type of problems considered are those where the diffusion coefficient is independent of the control variables, which are supposed to be interior to the control region. Two referees provided useful suggestions. Both authors gratefully acknowledge financial support from the regional Government of Castilla y León (Spain) under Project VA099/04, the Spanish Ministry of Education and Science and FEDER funds under Project MTM2005-06534.     

Stochastic Linear Quadratic Optimal Control Problems

This paper is concerned with the stochastic linear quadratic optimal control problem (LQ problem, for short) for which the coefficients are allowed to be random and the cost functional is allowed to have a negative weight on the square of the control variable. Some intrinsic relations among the LQ problem, the stochastic maximum principle, and the (linear) forward—backward stochastic differential equations are established. Some results involving Riccati equation are discussed as well. Accepted 15 May 2000. Online publication 1 December 2000     

参激白噪声作用下蔡电路的脉冲控制

考察了参激白噪声和脉冲信号联合作用下蔡电路的渐近P阶矩稳定性问题,得到该随机脉冲系统的比较系统,从而可由该确定性比较系统的稳定性得到原随机脉冲系统的渐近P阶矩稳定性.并从理论上得到能使该随机脉冲系统渐近P阶矩稳定的参数取值范围,即在稳定区域内取值的参数组合能够用脉冲方法对该随机蔡电路实现混沌控制.最后用数值仿真验证了理论结果的正确性.     

A Technique for Stochastic Control Problems with Unbounded Control Set

We describe a change of time technique for stochastic control problems with unbounded control set. We demonstrate the technique on a class of maximization problems that do not have optimal controls. Given such a problem, we introduce an extended problem which has the same value function as the original problem and for which there exist optimal controls that are expressible in simple terms. This device yields a natural sequence of suboptimal controls for the original problem. By this we mean a sequence of controls for which the payoff functions approach the value function.     

A Connection between Singular Stochastic Control and Optimal Stopping

We show that the value function of a singular stochastic control problem is equal to the integral of the value function of an associated optimal stopping problem. The connection is proved for a general class of diffusions using the method of viscosity solutions.