Convergence analysis in stochastic nonlinear systems: Asymptotic behaviors in extended time scales and large populations

Jian, Jiamin

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Convergence analysis in stochastic nonlinear systems: Asymptotic behaviors in extended time scales and large populations

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This thesis focuses on the convergence analysis of stochastic nonlinear systems. Specifically, it delves into the study of their asymptotic behavior, an area of paramount importance for comprehending the long-term dynamics of these systems and the systems with large populations. We concentrate on two key aspects: firstly, the examination of the turnpike property within the context of stochastic control problems, and secondly, the investigation of the convergence of N-player games towards their corresponding mean field games (MFG). Firstly, we investigate the asymptotic behavior of the systems with long-term dynamics and the convergence is with respect to the time horizon. In the first project, we examine the limiting behavior of a specific class of linear quadratic stochastic optimal control problems and their corresponding value functions as the time horizon approaches infinity. We establish the consistency between the cell problem in weak KAM theory and the static optimization problem from the perspective of the turnpike property. Moreover, we provide the connection between the cell problem and the ergodic cost problem, and then the classical turnpike property and the turnpike property in terms of the cost function are identified. Different from the first project, next we examine the system complexity. More precisely, we consider the convergence behavior of systems with large populations. MFG has become widely accepted as an approximation for the N-player games, especially when the number of players is large enough. A fundamental question that arises in this context concerns the convergence rate of this approximation. In the second project, we study the convergence rate of the N-player Linear-Quadratic-Gaussian (LQG) games with a Markov chain as the common noise towards its asymptotic MFG. By postulating a Markovian structure via two auxiliary processes for the first and second moments of the MFG equilibrium and applying the fixed point condition in MFG, we first provide the characterization of the equilibrium measure in MFG with a finite-dimensional Riccati system of ODEs. Additionally, with an explicit coupling of the optimal trajectory of the N-player game driven by N-dimensional Brownian motion and MFG counterpart driven by one-dimensional Brownian motion, we obtain the convergence rate 1/2 with respect to 2-Wasserstein distance. The number of states of the common noise considered in the above project is finite, thus it is natural to consider the case when the number of states of the common noise is infinity. In the third project, we focus on exploring the convergence properties of a generic player's trajectory and empirical measures in an N-player LQG Nash game, where Brownian motion serves as the common noise. The study establishes three distinct convergence rates concerning the representative player and empirical measure. To investigate the convergence, our methodology relies on a specific decomposition of the equilibrium path in the N-player game and utilizes the associated MFG framework. The basic structure of standard MFG theory assumes symmetry in the connections of the agents but not necessarily in their dynamics. However, asymmetric graph connections in large population games are considered in recent studies. In the network limit, a graphon gives the communication weights. In the last project, we consider the solvability of graphon mean field games. A new type of mean field games PDE system associated with the graphon mean field games is proposed. We establish the existence of solutions via the application of Schauder’s fixed point theorem and obtain the uniqueness of solution via the Lasry-Lions monotonicity assumption on the running cost.

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