Markov processes and applications

本建议的重点是概率论，专门研究马尔可夫过程及其应用。拟议研究计划的目标是在培训完成后，为最有可能和最理想的就业市场提供具有竞争力的研究组合，以达到HQP的稳定输出。培训分为三个层次；即博士后、研究生和本科生。目标是为博士后提供学术工作，而为研究生提供工业工作。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。***博士后项目的重点是超级进程及其变体。超级过程是作为数学工具出现的，用来模拟由于空间运动、种群成员的繁殖和死亡而发生变化的种群动态。本研究计划将专门研究这类种群的近边界行为，并将在给定其边界信息的情况下建立种群成员谱系的条件分布。从理论的角度来看，该研究计划的重要性是由于它与某一类非线性偏微分方程的分析以及无限维设置中的马丁边界理论的联系。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，以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征是，我们通过使用投资者的偏好结构将两者整合起来，共同建立了历史概率度量（控制破产概率）和风险中性度量（决定违约证券价格）的模型。从长远来看，该研究计划的目标是增加HQP培训机会，以更广泛地应用概率论，例如能源和数学生物学。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。本科课程的目的是双重的。首先，被培养为本科生的学生可能会被招募去攻读概率论的研究生学位。其次，通过小型和短期项目建立的专业知识可能会为研究生带来更大、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. **