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New Simulation Methods for Levy Processes and Related Distributions

Levy 过程和相关分布的新模拟方法

基本信息

批准号：
1720218
负责人：
Zhiyi Chi
金额：
$ 20.13万
依托单位：
University of Connecticut
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2021-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1720218&HistoricalAwards=false
关键词：
New Simulation Methods Levy Processes

项目摘要

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.

在高功率计算时代，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数值解决方案的关键。除了在金融和保险领域的普遍应用之外，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中也非常有用，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方法来推动 Levy 过程取得进展，重点关注高维情况。该项目的成果将被纳入课程材料中，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。封闭式分布函数的缺乏是随机抽样最严重的障碍之一。该项目建议建立在 PI 开发的所谓“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采样，并使用所谓的“泊松伽玛正态”近似对单变量无限可分分布进行近似采样。根据这些初步发现，该项目的目标是双重的。第一个目标是为多元过程开发新的随机抽样方法。其次，该项目将开发改进的或新的想法和方法来对单变量过程进行采样。为了实现这一目标，该项目将从两个方面研究随机抽样问题。首先，将为 Levy 过程的第一次通过事件或第一次退出事件开发精确的采样方法。这些随机事件在应用和理论上都发挥着重要作用，但目前，对其精确采样的了解非常有限，特别是对于多变量过程。其次，针对无限可分随机变量和Levy过程开发高精度近似采样方法，重点是易于实现。