Decentralized optimization and algorithms for stochastic dynamical systems with applications

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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.