Markov processes with applications to finance

马尔可夫过程是对许多具有随机行为的真实的生命系统进行数学建模的有力工具。 除了它们在这些系统中的应用外，对这一领域的研究还培育了数学和统计学的其他分支。 该研究计划的目标是马尔可夫过程理论中三个重要领域的发展。 第一个领域是应用于金融市场建模的随机微积分，重点是信贷市场系统。 我们的目标是开发概率模型，以了解的行为利率的债务发行的公司是受破产风险，以及量化的系统，其中有许多这样的公司相互作用的稳定性。 第二个领域是测量值过程，一个家庭的马尔可夫过程，研究开创了发展的人口遗传学模型。 该研究计划的目标是了解这些数学对象的更精细的性质，其中最重要的是超布朗运动。 特别是，我们有兴趣了解这个过程从域的退出行为。 事实证明，这也有重要的联系，另一个领域的数学，即研究偏微分方程。 第三个领域是马尔可夫链蒙特卡罗方法，这是强大的数值近似工具，特别是重要的贝叶斯估计。 我们的目标是得到这种近似方法的收敛速度的精确界。

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.