Markov processes and applications

这项建议的重点是专门研究马尔可夫过程及其应用的概率理论。拟议研究计划的目标是实现HQP的稳定产出，并配备有竞争力的研究组合，以便在完成培训后最有可能和最希望进入的就业市场进入。培训分为三个层次：博士后、研究生和本科生。目标是博士后研究员的学术工作，而研究生的工业工作。本科课程的目标是为概率及其应用领域的研究和发现提供一个切入点。 博士后课程的重点是超级过程及其变体。超过程作为一种数学工具应运而生，用于模拟由于种群成员的空间移动、繁殖和死亡而发生变化的种群的动态。这项研究计划将专门研究这类群体的接近边界行为，并将根据给定的群体成员的边界信息建立其谱系的条件分布。从理论上看，研究程序的重要性在于它与一类非线性偏微分方程的分析有关，也与无限维环境下的马丁边界理论有关。 该研究生课程专门研究数学金融的应用，特别是信用风险。我们的目标是建立数学模型，准确量化这一重要的风险来源，并评估受其影响的金融证券的公平价格。我们工作的一个显著特点是，通过使用投资者的偏好结构将历史概率指标和风险中性指标结合起来，我们联合建模了历史概率指标和风险中性指标，前者管理破产概率，后者决定可违约证券的价格。从长远来看，该研究计划的目标是为概率的更广泛应用增加HQP培训机会，例如在能源和数学生物学方面。 虽然本科课程的重点是广泛的，但即将到来的一年的主题将是风能。本科课程的目的有两个。首先，被培训为本科生的学生可能会被招募攻读概率论研究生学位。其次，通过小项目和短期项目积累的专业知识可能会为研究生带来更大和更长期的项目。

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.