Grant: Discovery Grants Program - Individual

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"". Levy processes are probably the most widely used objects in Applied Probability. They appear in such diverse areas as Mathematical Finance and Actuarial Mathematics, Queueing and Stochastic Control Theories, Physics and Chemistry. One particular problem that was well studied (and is now well understood) is the so-called ""exit problem"" for Levy processes, where we try to describe how the process exits the half-line or the finite interval. For example, we might be interested in the time when the process exits, the location before and after the exit, maximum and minimum of the process, etc. Exit problems in their different forms arise when one is pricing various exotic options in Mathematical Finance, when one is modeling the ruin event of an insurance company, etc. The main objective of the proposed research is to study the exit problems for Levy process in higher dimensions. This is a very interesting and important problem both from practical and theoretical points of view. For example, in the area of Mathematical Finance these results would be useful for describing the joint dynamics of several firms, or of a stock and an index, while in Actuarial Mathematics they would be useful when one is modeling the joint ruin event of several insurance companies, or of an insurance company and its reinsurer. Despite the importance of this problem, this area remains largely unexplored. We feel that the analytical approach which we have developed recently for one-dimensional Levy processes can be successfully generalized to higher dimensions and will lead us to many interesting results, at the same time opening new and exciting directions of research.