Markov processes with applications to finance

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Markov processes are powerful mathematical tools used in mathematical modeling of many real life systems with random behavior. Besides their applications to such systems, the study of this field nurtures other branches of mathematics as well as statistics. This research program targets developments in three important areas within the Markov process theory. The first area is stochastic calculus applied to the modeling of financial markets, with a focus on credit market systems. The goals are to develop probabilistic models for understanding the behavior of interest rates on debts issued by firms which are subject to bankruptcy risk as well as for quantifying stability properties of a system which has many such firms interacting with each other. The second area is measured valued processes, a family of Markov processes, the study of which pioneered the development of population genetics models. The goal of this research program is to understand finer properties of these mathematical objects, the most important one being Super-Brownian motion. In particular we are interested in understanding the exit behavior of this process from a domain. It turns out that this also has important connections to another area of mathematics, namely to the study of partial differential equations. The third area is Markov chain Monte Carlo methods, which are powerful numerical approximation tools especially important for Bayesian estimation. The goal is to get precise bounds on the rate of convergence of this approximation method.