Stochastic systems are systems in which random disturbances play a significant role. Stemming from emerging and existing applications in networked systems, wireless communications, signal processing, economics, and ecology, this project encompasses the study of dynamically evolving stochastic systems with uncertainties, switching among different configurations, and complex structures. The networked systems of interest include financial, communication, social, biological, and ecological networks. The work will be devoted to learning the intrinsic properties of stochastic network systems, developing mathematical models and novel mathematical methods for analyzing such systems, and designing efficient computational schemes for optimization and control of such systems to meet desired goals. The results of the research will be useful for applications to economics, nonlinear system identification and estimation, un-manned vehicles and other multi-agent systems, biodiversity in ecological systems, and social networks. This projects will involve undergraduate and graduate students and will integrate the research with teaching and student training. This work will contribute jointly to the further development of mathematical theory, computational methods and applications, and the improvement of mathematics education.
Motivated by a wide variety of applications, this project will study the following research topics. (1)Stochastic models with random switching will be developed and analyzed. Novel features of the systems include (i) past-dependent switching having a countable state space, and (ii) switching jump diffusions with non-local operators, finite switching set, and sigma finite jump measures. Criteria for recurrence, positive recurrence, and ergodicity will be obtained. (2) Kolmogorov-type systems under white noise perturbations, where the diffusions are degenerate, will be investigated. Applications to control dependent environmental protection zones, infectious disease and ecology will be studied. (3) New algorithms for switching diffusions and stochastic approximation will be developed and their rates of convergence will be studied. (i) For Milstein-type algorithms for solutions of switching diffusions, it will be shown that the algorithms preserve order 1 convergence rates as their diffusion counterpart. (ii) Motivated by applications to multi-agent systems, consensus, and swarming, the novelties of the stochastic approximation algorithms include the inclusion of state-dependent switching, state-dependent observation noise, and general time-dependent nonlinear functions. (4) Precise error estimates for identification of Hammerstein nonlinear systems with quantized observations will be obtained. It will be proved that the estimates escape from a small neighborhood of the true parameter with a probability that is exponentially small. (5) Accurate error bounds for approximation schemes of duplication-deletion random networks in the sense of strong approximation will be obtained. This will have impact on the study of random dynamic graphs and applications to social networks. Extensive numerical experiments and simulations will be performed to complement the analysis and algorithm design. The projects will involve the participation of undergraduate and graduate students.
