Complex Stochastic Systems: Analysis, Control and Applications

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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.