Markov processes and applications

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The focus of this proposal is probability theory specializing on Markov processes and their applications. The goal of the proposed research program is to reach a steady output of HQP equipped with a competitive research portfolio for the most likely and desired job market to enter upon the completion of the training. The training is stratified to three levels; namely, postdoctoral, graduate and undergraduate. The target is academic jobs for postdoctoral fellows, whereas industrial jobs for graduate students. The goal of the undergraduate program is to provide an entry point for research and discovery in the area of probability and its applications.***The postdoctoral program focuses on super-processes and their variants. Super-processes arise as mathematical tools to model the dynamics of populations that are subject to change due to the spatial movement, reproduction and death of the population members. This research program will specialize in near boundary behavior of such populations, and will build conditional distributions of the genealogies of the members of the population given their boundary information. From a theoretical point of view, the importance of the research program is due to its connection to the analysis of a certain type of non-linear partial differential equation and also to Martin boundary theory in an infinite dimensional setting.***The graduate program specializes in applications to mathematical finance, in particular, credit risk. Our goal is to create mathematical models to accurately quantify this important source of risk and assess the fair price of financial securities which are subject to it. One distinguishing feature of our work is that we jointly model the historical probability measure, which governs the bankruptcy probabilities, and the risk neutral measure, which determines the prices of defaultable securities, by integrating the two using investors' preference structures. In the long term, the research program will aim to increase HQP training opportunities for a wider range of applications of probability, for example to energy and mathematical biology.***While the intention for the undergraduate program's focus is to be broad, the theme for the upcoming year will be wind energy. The purpose of the undergraduate program is twofold. First the students who are trained as undergraduates may be recruited to pursue a graduate degree in probability theory. Secondly, the expertise built with small and short term projects may result in larger and longer term projects for graduate students. **