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: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=738837
• 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.

想象一个粒子以如下方式沿着直线运动：在每一时刻，粒子随机地（与过去无关）决定是向左还是向右跳。数学家称这个简单的模型为“离散时间随机漫步”。一个自然的推广这个模型的连续时间将被称为“一维列维过程”。丰富类列维过程占据了中心阶段，在大部分理论的随机过程。Levy过程在研究纯概率中许多重要对象的精细性质中是必不可少的，如分支过程、随机树、碎裂过程和自相似Markov过程。它们在许多应用概率模型中也是非常重要的，特别是在排队论和最优控制、数学金融和精算数学等领域。Levy过程从20世纪40年代开始被研究，在20世纪60年代-70年代初和90年代后期受到高度关注。然而，在这一领域仍有许多重要的未解决问题。其中一个问题是关于研究一个稳定过程如何从一个区间中退出（一个稳定过程是唯一自相似的利维过程：它的定律在时间和空间的同时缩放下保持不变）。在这一领域取得任何进展都将是一项重要成就，并将导致在适用稳定进程的许多领域取得进展。我打算使用最近的Wiener-Hopf分解矩阵的结果来研究这个问题，使用复杂的分析和概率方法的混合。当前活跃的另一个领域是广义伽马卷积（GGC）的研究--这是一类非常有用的分布，具有很多分析结构，包括许多应用中使用的分布（对数正态分布、威布尔分布、帕累托分布等）。在这里，我计划集中发展的数值方法与这类分布，特别是计算和近似的拉普拉斯变换这些随机变量的方法。我还计划研究GGC类的多维概括，并研究以这种方式出现的依赖结构和Copula，并将其应用于精算科学和数学金融。我未来研究的第三个方向将是用更简单但计算效率更高的过程来近似任意Levy过程。我将考虑具有有理变换跳跃的过程（这些是用于计算目的的最简单的过程），并将开发用这些过程近似任何Levy过程的算法。我相信，这项工作将是有用的所有从业者和应用数学家谁是使用利维过程建模的目的。