Decentralized optimization and algorithms for stochastic dynamical systems with applications

具有平均场相互作用的大种群随机系统出现在广泛的背景中，包括社会经济系统、工程（如无线网络、交通系统等）、生物系统。近十年来，平均场博弈论在统计物理思想的基础上得到了快速发展，为解决多主体动态竞争决策问题的维数问题提供了强有力的工具。这一领域不断吸引着世界范围内众多研究者的关注，不断发现新的理论成果，开辟新的应用领域。在平均场博弈设置中，本研究计划旨在开发随机经济增长理论的重要应用。相关的经典文献是Brock和Mirman（1972）针对离散时间情况和Merton（1975）针对连续时间情况提出的内生随机增长模型；这些著作构成了随机增长理论的基础，这是经济学中一个长期活跃的领域。本研究将主要采用连续时间建模，同时也考虑离散时间情况的某些方面。我们首先将用随机微分方程描述的默顿资本增长动力学推广到“相互作用粒子系统”的情况，并制定一个平均场博弈。这个广义系统用来描述从事某种生产活动的大量经济主体的竞争行为。我们特别感兴趣的是解决拥堵效应，或称为负外部性，其中总资本水平的增加降低了个体代理人的生产效率。根据平均场博弈的基本思想，我们将利用个体自身的运行信息和整体产生的一些可预测的宏观量来设计个体的策略，并进一步研究个体微观优化行为所导致的平均场的形成。要部署的数学机制来执行这个项目包括最优控制理论，偏微分方程，随机过程等。其方法和结果将引起应用数学家、经济学家、系统和控制理论家的兴趣。这个研究项目的另一部分将在概率设置下研究社会舆论动态。这是社会科学、经济学和统计物理学家非常感兴趣的领域。我们的主要兴趣是解决(i)随机不确定性，它发生在一个给定的代理获取他人意见的过程中，以及（ii）自由意志引起的噪音，也就是说，一个人的意见可能会由于人类心理而随机改变。我们将在这种嘈杂的环境中设计谨慎的意见学习算法，并研究在个体层面上由简单的意见更新规则导致的集体模式的形成。该研究将通过数学建模和分析为理解某些社会文化现象的模式形成提供新的见解。

Large-population stochastic systems with mean field interactions arise in a broad range of backgrounds including social-economic systems, engineering (such as wireless networks, traffic systems, etc.), biological systems. In the past decade mean field game theory has experienced rapid development based on ideas in statistical physics and provides powerful tools to deal with the curse of dimensionality in dynamic competitive decision problems with many agents. This area continues to attract the attention of many researchers worldwide, discovering new theoretical results and opening up new areas of applications.Within the mean field game setup, this research program aims to develop significant applications to stochastic economic growth theory. The related classic literature is the endogenous stochastic growth models introduced by Brock and Mirman (1972) for the discrete time case, and by Merton (1975) for the continuous time case; these works form the foundation of stochastic growth theory, a long active area in economics. This research will mainly adopt the continuous time modeling while also considering certain aspects of the discrete time case. We will first generalize Merton's capital growth dynamics, described by a stochastic differential equation, to an "interacting particle system" situation and formulate a mean field game. This generalized system is used to describe the competitive behavior of a large number of economic agents engaged in a certain type of production activity. We are particularly interested in addressing the congestion effect, or called negative externality, where the increase of the aggregate capital level decreases the production efficiency of individual agents. Following the basic idea of mean field games, we will design the strategy of individuals using its own operational information and some predictable macroscopic quantity generated by the whole population, and will further examine the formation of the mean field resulting from the microscopic optimizing behavior of individuals. The mathematical machinery to be deployed to carry out this project includes optimal control theory, partial differential equations, stochastic processes, among others. The methodology and results will be of interest to applied mathematicians, economists, and system and control theorists.Another part of this research program will study social opinion dynamics in a probabilistic setting. This area is of great interest to social science, economics and statistical physicists. Our main interest is to address (i) random uncertainties which occur during a given agent's acquisition of others' opinions and (ii) free will induced noise, that is, a person's opinion may have random shift possibly due to human phycology. We will devise cautious opinion learning algorithms in such noisy environments, and study the formation of collective patterns resulting from simple opinion updating rules at the individual level. This research will offer new insights for understanding the pattern formation of certain social and cultural phenomena via mathematical modeling and analysis.