Exit problems for Levy processes

想象一个粒子以如下方式沿线移动：在每个时刻，粒子随机地(与过去无关)决定是向左还是向右跳跃。数学家会把这个简单的模型称为“离散时间随机游走”。这一模型在连续时间的自然推广将被称为“一维利维过程”。 过程的丰富类Levy在许多随机过程理论中占据着中心地位。在研究许多重要的纯概率对象的精细性质时，Levy过程是不可缺少的，如分枝过程、随机树、碎裂过程和自相似马尔可夫过程。它们在许多应用的概率模型中也是非常重要的，特别是在排队论和最优控制、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区域的首次退出时间、过程在退出前后时刻的位置、过程的上确界/下确界等。自20世纪30-40年代引入Levy过程以来，人们对退出问题进行了深入的研究，然而近十年来，人们对这一领域的兴趣激增，主要是由于Levy过程的大量应用。 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)易于分析的过程和(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退出问题的新方法来克服上述三个障碍。这种方法的创新之处在于用复数分析、积分变换理论、数论等强大的分析技术补充了经典的概率方法。

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.