New Simulation Methods for Levy Processes and Related Distributions

University Of Connecticut, Storrs CT

Investigators

Abstract

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.