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. **

想象一个粒子以下列方式沿着这条线行进：在每个时刻，粒子随机地（并且与过去无关）决定是跳到左边还是右边。数学家将这个简单的模型称为“离散时间随机游走”。该模型对连续时间的自然推广将被称为“一维 Levy 过程”。丰富的过程类列维在许多随机过程理论中占据着中心舞台。 Levy过程对于纯概率中许多重要对象的精细性质的研究是不可或缺的，例如分支过程、随机树、碎片过程和自相似马尔可夫过程。它们在许多应用概率模型中也非常重要，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 年代就开始被研究，在 1960 年代至 1970 年代初以及 1990 年代末之后，人们对 Levy 过程的兴趣高涨。然而，该领域仍有许多重要的未解决问题。这些问题之一涉及研究稳定过程过程如何从区间退出（稳定过程是唯一自相似的 Levy 过程：其定律在时间和空间的同时缩放下保持不变）。在这一领域取得的任何进展都将是一项重要成就，并将导致应用稳定流程的许多领域取得进展。我打算利用矩阵维纳-霍普夫分解的最新结果，结合复杂分析和概率方法来研究这个问题。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，具有大量分析结构，并且包括应用中使用的许多分布（对数正态分布、威布尔分布、帕累托分布等）。在这里，我计划重点开发处理此类分布的数值方法，特别是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 类的多维概括，并研究以这种方式出现的依赖结构和联结函数，并将它们应用到精算科学和数学金融中。我未来研究的第三个方向是通过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），并将开发用于通过这些过程来近似任何 Levy 过程的算法。我相信这项工作对于所有使用 Levy 过程进行建模的从业者和应用数学家来说都是有用的。 **