New Simulation Methods for Levy Processes and Related Distributions

在高性能计算时代，Levy过程和无限可分分布的随机模拟或采样是找到这些领域中许多复杂问题的数值解的关键。除了在金融和保险领域的广泛应用外，Levy过程和无限可分分布在物理或社会科学、技术、工程和工业的广泛领域都非常有用，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来推进Levy过程，重点是高维情况。该项目的结果将被整合到课程材料中，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。缺乏封闭形式的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所谓的“嵌入和提取”的方法，由PI开发的一个大类的单变量Levy过程的第一个出口事件的精确采样和所谓的“泊松-伽马-正常”近似的单变量无限可分分布的近似采样。 根据这些初步调查结果，该项目有两个目标。第一个目标是为多变量过程开发新的随机抽样方法。 第二，本计画将发展改良或新的想法与方法，以针对单变量过程进行取样。 为了达到这个目标，本项目将从两个方面研究随机抽样问题。 首先，精确的抽样方法将开发的第一个通道事件或第一个退出事件的Levy过程。 这些随机事件在理论和应用上都有着重要的作用，但目前对它们的精确采样的认识非常有限，尤其是对于多变量过程。 第二，高精度的近似抽样方法将开发无限可分的随机变量以及Levy过程，重点是易于实现。

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.