Exit problems for Levy processes

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”。丰富的类列维过程占据了大部分的随机过程理论的中心阶段。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。它们在许多应用的概率模型中也是非常重要的，特别是在风险理论和最优控制、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）。典型的研究对象包括从一个区域的第一个退出时间，在退出之前和之后的时刻过程的位置，过程的上确界/下确界等。自从20世纪30 - 40年代Levy过程引入以来，退出问题一直被深入研究，然而在最近十年里，这一领域的兴趣激增，主要由Levy过程的大量应用驱动。在这一领域及其应用领域的进一步发展的一个主要障碍是缺乏（i）明确的结果，（ii）分析易处理的过程和（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。该方法的新奇在于用来自复分析、积分变换理论、数论等的强有力的分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.