Exit problems for Levy processes

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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.