Modeling, Analysis, Optimization, Computation, and Applications of Stochastic Systems

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This project aims to study stochastic systems in which random disturbances play a significant role. This research will encompass the study of the dynamic behavior of mathematical models and applications in areas of ecological and biological systems, wireless communication, financial engineering, networked systems, and systems in control engineering. The research will focus on model systems under the random influence, switching among different configurations, and complex structures. The results will provide an understanding of the fundamental properties and the basic features of such modeling systems. This project will provide training opportunities for graduate and undergraduate students. The research will promote diversity and inclusion, increase public scientific literacy, and enhance interdisciplinary collaborations and the STEM workforce.This project will encompass analysis and computation of several important topics from emerging and existing applications in networked systems, control engineering, optimization of systems, wireless communications, biology, ecology, economics, and social networks. (1) It aims to develop a new methodology for analyzing switching jump-diffusion type Kolmogorov systems. Novel features to be studied include non-local behavior due to the jumps, and uncertain environment modeling using random switching. Long-standing fundamental issues such as minimal conditions needed for persistence and extinction in population dynamics will be addressed. (2) Treating discontinuity in the iterates and non-smooth dynamics in the limits for stochastic approximation algorithms is vitally important. This project will focus on this issue from a new angle. Stochastic differential inclusion limits will be obtained and used to ascertain rates of convergence and to improve asymptotic efficiency for the first time. (3) Although nonlinear filtering is an area deemed to be well developed, computation remains to be the main challenge because of the infinite dimensionality. This project aims to develop a methodology based on machine learning and neural networks with a new approach using adaptive learning rate recursion, leading to potentially more efficient computational methods. (4) A key in numerically solving nonlinear stochastic differential equations is to treat high nonlinearity and numerical finite time explosion. This project will develop a class of algorithms to handle the problem. A novel idea will be the use of randomly generated growing truncation bounds. Convergence and rates of convergence will be developed. (5) In response to the urgent need to handle coupled equations in networks, this project will focus on the study of coupled switching jump diffusions. By using ideas from dynamic systems and coupling methods in probability, this project aims to obtain stability and stabilization with impact on networked systems. Extensive numerical experiments and simulations will be performed to complement the mathematical analysis and algorithm design. It will open a new domain for further research in mathematics with a broader range of applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.