Exit problems for Levy processes

Levy 进程的退出问题

• 批准号: 341233-2013
• 负责人: Kuznetsov, Alexey
• 金额: $ 1.38万
• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=572206
• 关键词: Exit; problems; Levy; processes

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 generalization 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 queueing theory and optimal control, mathematical finance and actuarial mathematics. Exit problems study how a stochastic process exits certain regions (a half-line, an interval, etc.). Typical objects of interest include the first exit time from a region, the location of the process in the moment immediately before and after the exit, the supremum/infimum of the process, etc. Exit problems have been intensively studied ever since the introduction of Levy processes in 1930s-1940s, however in the recent decade there has been a surge of interest in this area, mostly driven by numerous applications of Levy processes. A major obstacle for further development in this field and its areas of applications is the lack of (i) explicit results, (ii) analytically tractable processes and (iii) efficient numerical methods. My proposed research focuses on overcoming the above three obstacles by developing new methods for solving various exit problems. The novelty of the approach lies in supplementing the classical probabilistic methods with the powerful analytical techniques coming from complex analysis, theory of integral transforms, number theory, etc.

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。 退出问题研究随机过程如何退出某些区域（半线，间隔等）。感兴趣的典型对象包括从一个区域的第一个退出时间，在退出前后的此刻的位置，该过程的最高/临界。在1930年代至1940年代的过程中，在最近的十年中，这一领域引起了人们的兴趣，主要是由征收流程的众多应用所驱动的。 该领域及其应用领域进一步发展的主要障碍是缺乏（i）明确的结果，（ii）可分析的可拖动过程和（iii）有效的数值方法。我提出的研究重点是通过开发解决各种出口问题的新方法来克服以上三个障碍。该方法的新颖性在于用来自复杂分析的强大分析技术，整体变换的理论，数字理论等来补充经典的概率方法。