This research project considers a class of complex stochastic systems and related stochastic control problems. The underlying systems are subject to various random forces. The specific aims and anticipated results of this project are as follows: (1) to investigate stability and ergodicity and to establish a Feynman-Kac type formula for regime-switching diffusions with jumps; (2) to develop singular stochastic control theories for regime-switching jump diffusions and to design feasible and effective numerical schemes for the associated control problems; and (3) to apply the theoretical results to biology, mathematical finance, and risk management. The expected results of this project will contribute to an in-depth understanding of a wide class of complex stochastic systems. This, in turn, will facilitate the applications of such systems in areas such as finance and biology. The mixed regular and singular control problems for regime-switching diffusions are likely to generate many new and interesting mathematical results as well as new problems in stochastic analysis, numerical approximation and control theory.
This research project is motivated by emerging applications arising from ecosystem modeling, financial engineering, insurance risk processes, manufacturing and production planning. The dynamics of these systems inevitably involve uncertainty. For example, in ecosystem modeling, the population dynamics of a general ecosystem possess two salient features: (i) there is day-to-day jitter that causes minor fluctuations as well as big population loss caused by rare events such as epidemics, earthquakes, and tsunamis; and (ii) there are qualitative changes in the system stemming from the fact that the growth rates and carrying capacities of many species often vary according to changes in nutrition, water supply, and/or food resources. Similar phenomena are observed in the dynamics of insurance risk processes, the price of a risky asset, and others. These features make the usual models in the literature inadequate in describing such complex systems. The proposed project aims to take into these inherent random forces and propose stochastic processes and related control problems that are general and flexible, yet mathematically tractable, in dealing with these real-world applications. It presents novel stochastic processes for modeling and analysis of complex systems, obtains long-time behavior of such systems, develops singular control theories, and designs numerical schemes for the control problems. Student training and education, disciplinary and interdisciplinary collaborations, and the dissemination of research results through publications and presentations are integral parts of this project.
The support of the NSF grant has helped the PI to be productive in the last three years. In the last three years, the PI has published four refereed journal papers, two invited book chapters, and six conference proceedings. In addition, the PI has four papers submitted to academic journals. These publications and preprints address different questions and goals of the proposed topics in the project. They develop singular control theories and related numerical approximation schemes for singular control problems for regime-switching diffusions, investigate asymptotic properties for regime-switching jump diffusions, establish several versions of the celebrated Feynman-Kac formula for regime-switching jump diffusions, and explore applications of such processes in risk management and mathematical finance. They contribute to a deep understanding of regime-switching jump diffusions. These, in turn, facilitate the applications of such complex stochastic processes in areas such as mathematical biology, risk management and mathematical finance. We have accomplished the major research goals of the project. The findings of the research project have been disseminated to a broad audience. In addition to the publications, the PI delivered 17 invited talks in international and national conferences and universities in and out of the US. The NSF support also helped the PI to continue and/or initiate collaborations with researchers around the world. The PI collaborates with colleagues of the Department of Mathematical Sciences, University of Wisconsin-Milwaukee, as well as with researchers from Australia, China, Germany, Hong Kong, and Ireland. The grant also supported the PI to visit Hong Kong and Beijing for research collaborations. Also, the NSF grant partially supported a research collaborator from Beijing Institute of Technology to visit Milwaukee in July 2014. During the visit, the PI and the collaborator initiated and made significant progress on a project on asymptotic properties of regime-switching jump diffusion processes with an infinite state space for the switching component. The findings of the project also helped the PI to supervise master's and Ph.D.'s theses. In fact, the theses topics were closely related to the proposed topics in the project. The master's thesis is on risk indifference pricing in incomplete markets modeled by regime-switching jump diffusions while the Ph.D.'s thesis focused on high order weak numerical solution to stochastic differential equations. In addition, the PI had the opportunity to supervise two undergraduate students' capstone projects in 2011. The PI also served in numerous masters' and Ph.D.s' theses defense committees. Part of the research findings of the project were used to enrich the graduate level course "Advanced Topics in Probability'' in the Spring 2012 and Fall 2013 semesters. In particular, the PI introduced topics such as singular control and Feynman-Kac formula for regime-switching jump diffusions in the course. In addition, the PI reported the findings and progresses of the research project on a regular basis in the probability seminar in the Department of Mathematical Sciences, UW-Milwaukee. The NSF support also helped the PI to develop interdisciplinary training opportunities for graduate students. The PI had the opportunity to work with civil engineers in the last two years on a project on stochastic modeling of the effect of FRP stiffness Variation on FRP debonding from concrete. The project involved faculty members from Departments of Civil, Construction and Environmental Engineering in Marquette University and Hong Kong Polytechnic University, as well as a female graduate student of Marquette University. The PI served in the Ph.D. proposal and defense committees for the female student. Part of the findings of the project were published in a conference proceeding in 2012. To summarize, we have successfully finished the project. Moreover, the project makes broad impacts in the integration of research and education.