Markov processes with applications to finance

马尔可夫过程是一种强大的数学工具，用于对许多具有随机行为的实际系统进行数学建模。除了它们在这类系统中的应用之外，这一领域的研究还培养了数学和统计学的其他分支。这个研究项目的目标是马尔可夫过程理论中三个重要领域的发展。第一个领域是随机微积分应用于金融市场的建模，重点是信贷市场系统。我们的目标是开发概率模型，用于理解受破产风险影响的公司发行的债务利率的行为，以及量化具有许多此类公司相互作用的系统的稳定性属性。第二个领域是测量价值过程，这是一族马尔可夫过程，对它的研究开创了种群遗传学模型的发展。这个研究项目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，我们对了解此流程从域的退出行为感兴趣。事实证明，这与另一个数学领域也有重要的联系，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，这是一种强大的数值逼近工具，对于贝叶斯估计特别重要。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.