Focusing on new algorithms design for stochastic optimization and investigating basic properties of systems arising in emerging applications, this research project encompasses the following four aspects. (1) It aims to develop stochastic approximation algorithms with Markovian switching and examining Markov modulated random sequences; it reveals asymptotic properties such as limit switching diffusions, large deviations, strong invariance, and ergodicity. (2) It presents numerical methods for solutions of regime-switching stochastic differential equations with continuous-state-depend switching. In addition to convergence, in the second phase, rates of convergence of the algorithms, numerical methods for the related control problems, and their convergence rates are to be investigated. (3) It carries out system identification with Markov parameter and binary-valued or quantized observations. It facilitates the understanding of tractability, complexity, and modeling capability under limited sensor information. (4) It analyzes stability of switching jump diffusions with state-dependent switching, and provides sufficient conditions for stability and instability of nonlinear systems and necessary and sufficient conditions for linearizable systems. Consisting of in-depth analysis and extensive numerical experiments, our goals are to gain new insight, and to advance state of the art of stochastic optimization methods and stochastic systems theory.
This research project is motivated by emerging applications arising in wireless communication, adaptive signal processing, production planning, queueing systems, biological, ecological, and economic systems, which are inevitably involve uncertainty. In contrast to the usual models in the existing literature, the systems are often influenced by random environment as well. For example, when two or more species live in proximity and share the same basic requirements, they usually compete for resources, food, habitat, or territory. Traditional models use (either random or non-random) differential equations for such scenarios. However, the systems are often subject to additional environmental noise, which cannot be described by the traditional differential equation setup. Other examples include insurance risk models and ion channel (biological nanotubes) dynamics among others. The proposed project aims to take into such random environment and other uncertain factors into consideration. It presents novel algorithms for optimization tasks, designs numerical procedures for solving systems of equations, carries out identification task for systems with unknown parameters and limited sensor information, and obtains longtime behavior of systems involving both continuous dynamics and discrete events. The models to be examined, the numerical algorithms to be developed, and the insight to be gained will jointly contribute to the field of stochastic optimization and make impact on the aforementioned applications. Several graduate students are involved in the research project. By integrating the proposed research with teaching, the planned work contributes to the further development of stochastic optimization and stochastic systems theory and the improvement of mathematics education.
