Exit problems for Levy processes

想象一个粒子以以下方式沿着这条线运动：在每一个时刻，粒子随机决定（并且独立于过去）是向左还是向右跳。数学家将这种简单的模型称为“离散时间随机漫步”。将这个模型自然地推广到连续时间将被称为“一维列维过程”。在随机过程的许多理论中，富有阶层的列维占有中心地位。在纯概率中研究分支过程、随机树、碎片过程和自相似马尔可夫过程等许多重要对象的精细性质时，Levy过程是必不可少的。它们在许多应用概率模型中也非常重要，特别是在排队理论和最优控制、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、区间等）。感兴趣的典型对象包括从一个区域的第一次退出时间，在退出之前和之后的时刻进程的位置，进程的最大值/最小值，等等。自20世纪30年代至40年代Levy过程引入以来，出口问题已经得到了深入研究，然而在最近十年中，在这一领域的兴趣激增，主要是由Levy过程的大量应用驱动的。这一领域及其应用领域进一步发展的一个主要障碍是缺乏(i)明确的结果，（ii）可分析处理的过程和（iii）有效的数值方法。我建议的研究重点是通过开发解决各种退出问题的新方法来克服上述三个障碍。该方法的新颖之处在于利用复分析、积分变换理论、数论等强大的解析技术来补充经典的概率方法。

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.