Summary We begin by introducing the important concepts of filtration, martingale and stopping time. These are then applied to establish the strong Markov property for Lévy processes and to prove that every Lévy process has a càdlàg modification. We then meet random measures, particularly those of Poisson type, and the associated Poisson integrals, which track the jumps of a Lévy process. The most important result of this chapter is the Lévy–Itôo decomposition of a Lévy process into a Brownian motion with drift (the continuous part), a Poisson integral (the large jumps) and a compensated Poisson integral (the small jumps). As a corollary, we complete the proof of the Lévy–Khintchine formula. Finally, we establish the interlacing construction, whereby a Lévy process is obtained as the almost-sure limit of a sequence of Brownian motions with drift wherein random jump discontinuities are inserted at random times.

In this chapter we will frequently encounter stochastic processes with càdlàg paths (i.e. paths that are continuous on the right and always have limits on the left). Readers requiring background knowledge in this area should consult the appendix at the end of the chapter.

Before you start reading this chapter, be aware that parts of it are quite technical. If you are mainly interested in applications, feel free to skim it, taking note of the results of the main theorems without worrying too much about the proofs.

Summary We begin this chapter by studying two different types of ‘exponential’ of a Lévy-type stochastic integral Y. The first of these is the stochastic exponential, dZ(t) = Z(t −)dY(t), and the second is the process eY. We are particularly interested in identifying conditions under which eY is a martingale. It can then be used to implement a change to an equivalent measure. This leads to Girsanov's theorem, and an important special case of this is the Cameron–Martin–Maruyama theorem, which underlies analysis in Wiener space. In Section 5.3 we prove the martingale representation theorem in the Brownian case and also discuss extensions to include jump processes. The final section of this chapter briefly surveys some applications to option pricing. We discuss the search for equivalent risk-neutral measures within a general ‘geometric Lévy process’ stock price model. In the Brownian case, we derive the Black–Scholes pricing formula for a European option. In the general case, where the market is incomplete, we discuss the Föllmer–Schweitzer minimal measure and Esscher transform approaches. The case where the market is driven by a hyperbolic Lévy process is discussed in some detail.

In this chapter, we will explore further important properties of stochastic integrals, particularly the implications of Itô's formula. Many of the developments which we will study here, although of considerable theoretical interest in their own right, are also essential tools in mathematical finance as we will see in the final section of this chapter.

The aim of this book is to provide a straightforward and accessible introduction to stochastic integrals and stochastic differential equations driven by Lévy processes.

Lévy processes are essentially stochastic processes with stationary and independent increments. Their importance in probability theory stems from the following facts:

• they are analogues of random walks in continuous time;

• they form special subclasses of both semimartingales and Markov processes for which the analysis is on the one hand much simpler and on the other hand provides valuable guidance for the general case;

• they are the simplest examples of random motion whose sample paths are right-continuous and have a number (at most countable) of random jump discontinuities occurring at random times, on each finite time interval.

• they include a number of very important processes as special cases, including Brownian motion, the Poisson process, stable and self-decomposable processes and subordinators.

Although much of the basic theory was established in the 1930s, recent years have seen a great deal of new theoretical development as well as novel applications in such diverse areas as mathematical finance and quantum field theory. Recent texts that have given systematic expositions of the theory have been Bertoin and Sato. Samorodnitsky and Taqqu is a bible for stable processes and related ideas of self-similarity, while a more applications-oriented view of the stable world can be found in Uchaikin and Zolotarev.

We define a stochastic flow on a manifold M as a right Lévy process in Diff(M). In this chapter, we look at the dynamical aspects of a right Lévy process regarded as a stochastic flow. The first section contains some basic definitions and facts about a general stochastic flow. Although these facts will not be used to prove anything, they provide a general setting under which one may gain a better understanding of the results to be proved. In the rest of the chapter, the limiting properties of Lévy processes are applied to study the asymptotic stability of the induced stochastic flows on certain compact homogeneous spaces. In Section 8.2, the properties of the Lévy process are transformed to a form more suitable for the study of its dynamical behavior, in which the dependence on ω and the initial point g is made explicit. In Section 8.3, the explicit formulas, in terms of the group structure, for the Lyapunov exponents and the associated stable manifolds are obtained. A clustering property of the stochastic flow related to the rate vector of the Lévy process is studied in Section 8.4. Some explicit results for SL(d, ℝ)-flows and SO(1, d)-flows on SO(d) and Sd−1 are presented in the last three sections. The main results of this chapter are taken from Liao [40, 41, 42].

Like the simple additive structure on an Euclidean space, the more complicated algebraic structure on a Lie group provides a convenient setting under which various stochastic processes with interesting properties may be defined and studied. An important class of such processes are Lévy processes that possess translation invariant distributions. Since a Lie group is in general noncommutative, there are two different types of Lévy processes, left and right Lévy processes, defined respectively by the left and right translations. Because the two are in natural duality, for most purposes, it suffices to study only one of them and derive the results for the other process by a simple transformation. However, the two processes play different roles in applications. Note that a Lévy process may also be characterized as a process that possesses independent and stationary increments.

The theory of Lévy processes in Lie groups is not merely an extension of the theory of Lévy processes in Euclidean spaces. Because of the unique structures possessed by the noncommutative Lie groups, these processes exhibit certain interesting properties that are not present for their counterparts in Euclidean spaces. These properties reveal a deep connection between the behavior of the stochastic processes and the underlying algebraic and geometric structures of the Lie groups.

In this chapter, we consider the processes in a homogeneous space of a Lie group G induced by Lévy processes in G. In Section 2.1, these processes are introduced as one-point motions of Lévy processes in Lie groups. We derive the stochastic integral equations satisfied by these processes and discuss their Markov property. In Section 2.2, we consider the Markov processes in a homogeneous space of G that are invariant under the action of G. We study the relations among various invariance properties, we present Hunt's result on the generators of G-invariant processes, and we show that these processes are one-point motions of left Lévy processes in G that are also invariant under the right action of the isotropy subgroup. The last section of this chapter contains a discussion of Riemannian Brownian motions in Lie groups and homogeneous spaces.

One-Point Motions

Let G be a Lie group that acts on a manifold M on the left and let gt be a process in G. For any x ∈ M, the process xt = gtx will be called the one-point motion of gt in M starting from x. In general, the one-point motion of a Markov process in G is not a Markov process in M.

This chapter contains an introduction to Lévy processes in a general Lie group. The left and right Lévy processes in a topological group G are defined in Section 1.1. They can be constructed from a convolution semigroup of probability measures on G and are Markov processes with left or right invariant Feller transition semigroups. In the next two sections, we introduce Hunt's theorem for the generator of a Lévy process in a Lie group G and prove some related results for the Lévy measure determined by the jumps of the process. In Section 1.4, the Lévy process is characterized as a solution of a stochastic integral equation driven by a Brownian motion and an independent Poisson random measure whose characteristic measure is the Lévy measure. Some variations and extensions of this stochastic integral equation are discussed. The proofs of the stochastic integral equation characterization, due to Applebaum and Kunita, and of Hunt's theorem, will be given in Chapter 3. For Lévy processes in matrix groups, a more explicit stochastic integral equation, written in matrix form, is obtained in Section 1.5.

Lévy Processes

The reader is referred to Appendices A and B for the basic definitions and facts on Lie groups, stochastic processes, and stochastic analysis.

The present volume provides an introduction to Lévy processes in general Lie groups, and hopefully an accessible account on the limiting and dynamical properties of such processes in semi-simple Lie groups of non-compact type. Lévy processes in Euclidean spaces, including the famous Brownian motion, have always played a central role in probability theory. In recent times, there has been intense research activity in exploring the probabilistic connections of various algebraic and geometric structures, therefore, the study of stochastic processes in Lie groups has become increasingly important. This book is aimed at serving two purposes. First, it may provide a foundation to the theory of Lévy processes in Lie groups, as this is perhaps the first book written on the subject, and second it will present some important results in this area, revealing an interesting connection between probability and Lie groups.

Please note: when referring to a result in a referenced text, the chapter and enunciation numbering system in that text has been followed.

David Applebaum, Olav Kallenberg and Wang Longmin read portions of the manuscript and provided useful comments. Part of the book was written during the author's visit to Nankai University, Tianjin, China in the fall of 2002. I wish to take this opportunity to thank my hosts, Wu Rong and Zhou Xingwei, for their hospitality. It would be hard to imagine this work ever being completed without my wife's support and understanding.

In this chapter, we apply Fourier analysis to study the distributions of Lévy processes gt in compact Lie groups. After a brief review of the Fourier analysis on compact Lie groups based on the Peter—Weyl theorem, we discuss in Section 4.2 the Fourier expansion of the distribution density pt of a Lévy process gt in terms of matrix elements of irreducible unitary representations of G. It is shown that if gt has an L2 density pt, then the Fourier series converges absolutely and uniformly on G, and the coefficients tend to 0 exponentially as time t → ∞. In Section 4.3, for Lévy processes invariant under the inverse map, the L2 distribution density is shown to exist, and the exponential bounds for the density as well as the exponential convergence of the distribution to the normalized Haar measure are obtained. The same results are proved in Section 4.4 for conjugate invariant Lévy processes. In this case, the Fourier expansion is given in terms of irreducible characters, a more manageable form of Fourier series. An example on the special unitary group SU(2) is computed explicitly in the last section. The results of this chapter are taken from Liao [43].

Fourier Analysis on Compact Lie Groups

This section is devoted to a brief discussion of Fourier series of L2 functions on a compact Lie group G based on the Peter—Weyl theorem.