New Simulation Methods for Levy Processes and Related Distributions

在高功率计算的时代，Levy过程和无限可分分布的随机模拟或抽样是找到这些领域许多复杂问题的数值解的关键。除了在金融和保险中的普遍应用外，Levy过程和无穷可分分布在物理或社会科学、技术、工程和工业的广泛领域中都非常有用，例如湍流、激光冷却、互联网流量、通信或服务器队列、设备维护、健康危害监测和临床试验。该项目旨在通过开发新的方法，重点放在高维度的情况下，在Levy过程中取得进展。该项目的成果将被整合到教材中，以吸引不同背景的学生研究随机抽样及其在其他领域的应用。缺乏闭式分布函数是随机抽样最严重的障碍之一。这个项目建议建立在PI开发的所谓的“嵌入和提取”方法的基础上，对一大类单变量Levy过程的第一个退出事件进行精确抽样，并对单变量无穷可分分布的近似抽样建立所谓的“泊松-伽马-正态”近似。基于这些初步发现，该项目的目标是双重的。第一个目标是为多变量过程发展新的随机抽样方法。其次，该项目将开发改进的或新的想法和方法来为单变量过程采样。为了实现这一目标，该项目将从两个方面对随机抽样问题进行调查。首先，将为Levy过程的首次通过事件或首次退出事件制定准确的抽样方法。这些随机事件无论在理论上还是在应用上都有着重要的作用，但目前关于它们的精确抽样的知识非常有限，尤其是对于多变量过程。其次，将为无限可除随机变量和Levy过程开发高精度的近似抽样方法，并强调易于实施。

In the age of high power computing, random simulation, or sampling, of Levy processes and infinitely divisible distributions is key to finding numerical solutions to many complicated problems in those fields. In addition to their ubiquitous application in finance and insurance, Levy processes and infinitely divisible distributions are extremely useful in a wide range of fields of physical or social science, technology, engineering, and industry, such as turbulence, laser cooling, internet traffic, communication or server queues, equipment maintenance, health hazard monitoring, and clinical trial. The project aims to make advances in Levy processes through development of novel approaches, with emphasis on high dimensional situations. The results of the project will be integrated into course material to attract students with diverse backgrounds to research on random sampling and its applications in other fields.The lack of closed-form distribution functions is one of the most serious hurdles to the random sampling. This project proposes to build on the so-called "embed-and-extract" approach developed by the PI to exact sampling of the first exit event of a large class of univariate Levy processes and a so-called "Poisson-gamma-normal" approximation for approximate sampling of univariate infinitely divisible distributions. Based on these preliminary findings, the goal of the project is twofold. The first goal is to develop new random sampling methods for the multivariate processes. Second, the project will develop refined or new ideas and methods to sample for the univariate processes. To achieve the goal, the project will investigate the issue of random sampling from two aspects. First, exact sampling methods will be developed for the first passage event or first exit event of a Levy process. These random events play an important role in applications as well as in theory, but at present, the knowledge on their exact sampling is very limited, especially for multivariate processes. Second, high-precision approximate sampling methods will be developed for infinitely divisible random variables as well as Levy processes, with the emphasis on easy implementation.