Levy processes and their applications

征收流程及其应用

• 批准号: RGPIN-2019-06320
• 负责人: Kuznetsov, Alexey
• 金额: $ 1.82万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=680846
• 关键词: Levy; processes; their; applications

Imagine a particle that travels along the line in the following way: At each moment of time the particle decides randomly (and independently of the past) whether to jump to the left or to the right. Mathematicians would call this simple model a "discrete time random walk". A natural generalisation of this model to continuous time would be called a "one-dimensional Levy process". The rich class Levy of processes occupies the central stage in much of the theory of stochastic processes. Levy processes are indispensable in the study of fine properties of many important objects in pure probability, such as branching processes, random trees, fragmentation processes and self-similar Markov processes. They are also all-important in many applied probability models, in particular in such areas as queuing theory and optimal control, mathematical finance and actuarial mathematics.***Levy processes have been studied from 1940s, with periods of heightened interest in 1960s- early 1970s and after late 1990s. However, there are still many important unresolved problems in this area. One of these problems concerns investigating how a stable process process exits from an interval (a stable process is the only Levy process that is self-similar: its law is preserved under a simultaneous scaling of time and space). Making any progress in this area would be an important achievement and would lead to advances in many areas where stable processes are applied. I intend to use recent results on Wiener-Hopf factorization for matrices to study this problem using a mix of complex-analytical and probabilistic methods. Another area of intense current activity is the study of Generalized Gamma Convolutions (GGC) -- a very useful class of distributions that has a lot of analytical structure and that includes many distributions used in applications (lognormal, Weibull, Pareto, etc.). Here I plan to focus on developing numerical methods for working with this class of distributions, in particular, methods for computing and approximating Laplace transforms of these random variables. I also plan to investigate multi-dimensional generalisations of the GGC class and to study dependence structures and copulas that arise in this way and apply them in Actuarial Science and Mathematical Finance. The third direction of my future research will be about approximating arbitrary Levy processes by simpler, but more computationally efficient processes. I will consider processes with jumps of rational transform (these are the easiest processes for computational purposes) and will develop algorithms for approximating any Levy process by these ones. I believe that this work will be useful for all practitioners and applied mathematicians who are using Levy processes for modelling purposes. **

想象一个粒子以下面的方式沿着线路传播：在每时刻，粒子都会随机决定（并且独立于过去）是否跳到左或向右跳。数学家将此简单模型称为“离散时间随机步行”。该模型对连续时间的自然概括将被称为“一维征费过程”。在随机过程的大部分理论中，丰富的过程征收均占据了中心阶段。在研究许多重要对象在纯概率中的精细特性的研究中，征费过程是必不可少的，例如分支过程，随机树，破碎过程和自相似的马尔可夫过程。它们在许多应用概率模型中也很重要，尤其是在排队理论和最佳控制，数学金融和精算数学等领域。但是，该领域仍然存在许多重要的未解决问题。这些问题之一涉及研究稳定过程如何从间隔中退出的问题（稳定过程是唯一具有相似性的征税过程：其定律是在同时扩大时间和空间下保存的）。在这一领域取得任何进展将是一项重要的成就，并将在应用稳定过程的许多领域取得进步。我打算使用Wiener-HOPF分解的最新结果进行矩阵，以使用复杂的分析和概率方法的混合来研究此问题。当前活动的另一个领域是对广义γ卷积（GGC）的研究 - 一种非常有用的分布类别具有大量的分析结构，其中包括应用中使用的许多分布（lognormal，weibull，weibull，Pareto等）。在这里，我计划专注于开发用于使用此类分布的数值方法，特别是用于计算和近似这些随机变量的拉普拉斯变换的方法。我还计划研究GGC类别的多维概括，并研究以这种方式出现的依赖性结构和Copulas，并将其应用于精算科学和数学金融中。我未来的研究的第三个方向将是通过更简单但更有效的过程来近似任意征税过程。我将考虑使用理性转换的跳跃（这些是用于计算目的的最简单过程）的过程，并将开发算法以近似这些征税过程。我认为，这项工作对于使用征费过程进行建模目的的所有从业者和应用数学家都会有用。 **