Levy processes and their applications

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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 generalisation 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 queuing theory and optimal control, mathematical finance and actuarial mathematics.***Levy processes have been studied from 1940s, with periods of heightened interest in 1960s- early 1970s and after late 1990s. However, there are still many important unresolved problems in this area. One of these problems concerns investigating how a stable process process exits from an interval (a stable process is the only Levy process that is self-similar: its law is preserved under a simultaneous scaling of time and space). Making any progress in this area would be an important achievement and would lead to advances in many areas where stable processes are applied. I intend to use recent results on Wiener-Hopf factorization for matrices to study this problem using a mix of complex-analytical and probabilistic methods. Another area of intense current activity is the study of Generalized Gamma Convolutions (GGC) -- a very useful class of distributions that has a lot of analytical structure and that includes many distributions used in applications (lognormal, Weibull, Pareto, etc.). Here I plan to focus on developing numerical methods for working with this class of distributions, in particular, methods for computing and approximating Laplace transforms of these random variables. I also plan to investigate multi-dimensional generalisations of the GGC class and to study dependence structures and copulas that arise in this way and apply them in Actuarial Science and Mathematical Finance. The third direction of my future research will be about approximating arbitrary Levy processes by simpler, but more computationally efficient processes. I will consider processes with jumps of rational transform (these are the easiest processes for computational purposes) and will develop algorithms for approximating any Levy process by these ones. I believe that this work will be useful for all practitioners and applied mathematicians who are using Levy processes for modelling purposes. **