Lévy Processes — From Probability to Finance and Quantum Groups

1320 NOTICES OF THE AMS VOLUME 51, NUMBER 11 T he theory of stochastic processes was one of the most important mathematical developments of the twentieth century. Intuitively, it aims to model the interaction of “chance” with “time”. The tools with which this is made precise were provided by the great Russian mathematician A. N. Kolmogorov in the 1930s. He realized that probability can be rigorously founded on measure theory, and then a stochastic process is a family of random variables (X(t), t ≥ 0) defined on a probability space (Ω,F , P ) and taking values in a measurable space (E,E) . Here Ω is a set (the sample space of possible outcomes), F is a σ-algebra of subsets of Ω (the events), and P is a positive measure of total mass 1 on (Ω,F ) (the probability). E is sometimes called the state space. Each X(t) is a (F ,E) measurable mapping from Ω to E and should be thought of as a random observation made on E made at time t . For many developments, both theoretical and applied, E is Euclidean space Rd (often with d = 1); however, there is also considerable interest in the case where E is an infinite dimensional Hilbert or Banach space, or a finite-dimensional Lie group or manifold. In all of these cases E can be taken to be the Borel σalgebra generated by the open sets. To model probabilities arising within quantum theory, the scheme described above is insufficiently general and must be embedded into a suitable noncommutative structure. Stochastic processes are not only mathematically rich objects. They also have an extensive range of applications in, e.g., physics, engineering, ecology, and economics—indeed, it is difficult to conceive of a quantitative discipline in which they do not feature. There is a limited amount that can be said about the general concept, and much of both theory and applications focusses on the properties of specific classes of process that possess additional structure. Many of these, such as random walks and Markov chains, will be well known to readers. Others, such as semimartingales and measure-valued diffusions, are more esoteric. In this article, I will give an introduction to a class of stochastic processes called Lévy processes, in honor of the great French probabilist Paul Lévy, who first studied them in the 1930s. Their basic structure was understood during the “heroic age” of probability in the 1930s and 1940s and much of this was due to Paul Lévy himself, the Russian mathematician A. N. Khintchine, and to K. Itô in Japan. During the past ten years, there has been a great revival of interest in these processes, due to new theoretical developments and also a wealth of novel applications—particularly to option pricing in mathematical finance. As well as a vast number of research papers, a number of books on the subject have been published ([3], [11], [1], [2], [12]) and there have been annual international conferences devoted to these processes since 1998. Before we begin the main part of the article, it is worth David Applebaum is professor of probability and statistics at the University of Sheffield. His email address is D.Applebaum@sheffield.ac.uk. He is the author of Lévy Processes and Stochastic Calculus, Cambridge University Press, 2004, on which part of this article is based.