This book has germinated from the lecture notes of a course ‘Large deviations and metastability’ given by one of us at the ‘CIMPA First School on Dynamical and Disordered Systems’, at Universidad de la Frontera, Temuco, during the summer of 1992 [293].

Since then a large amount of new material on metastability has been accumulated, and our goal was to combine a basic introduction to the theory of large deviations with a wide overview of the metastable behaviour of stochastic dynamics.

Typical examples of metastable states are supersaturated vapours and magnetic systems with magnetization opposite to the external field. Metastable behaviour is characterized by a long period of apparent equilibrium of a pure thermodynamic phase followed by an unexpected fast decay towards the stable equilibrium of a different pure phase or of a mixture, e.g. homogeneous nucleation of the liquid phase inside a highly supersaturated vapour, due to spontaneous density fluctuations. The point of view of metastability as a genuinely dynamical phenomenon is now widely accepted. Approaches which aim to describe static aspects of metastability (such as determination of the metastable branch of the equation of state of a fluid) in the Gibbs equilibrium set-up are, in their ‘naïve form’, applicable only in a mean field context. In this case, the physically unacceptable assumption that the range of the interaction equals the linear dimension of the container gives rise to pathological behaviour of non-convex free energy that implies negative compressibility, namely, thermodynamic instability.

The van der Waals–Maxwell theory

Metastability is a relevant phenomenon for thermodynamic systems close to a first order phase transition. Examples are supercooled vapours and liquids, super-saturated vapours and solutions, as well as ferromagnets in the part of the hysteresis loop where the magnetization is opposite to the external magnetic field. A metastable state occurs when some thermodynamic parameter such as the temperature, pressure or magnetic field is changed from a value giving rise to a stable state with a unique phase, say X, to one for which at least part of the system should be in some new equilibrium phase Y. Then, in particular experimental situations, instead of undergoing the phase transition, the system goes over continuously into a ‘false’ equilibrium state with a unique phase X′, far from Y but actually close to the initial equilibrium phase X. It is this apparent equilibrium situation that is called a ‘metastable state’. Its properties are very similar to those of the stable equilibrium state; for example for a supersaturated vapour one can determine the pressure experimentally as a function of the temperature and the specific volume. We speak of the ‘metastable branch’ of the isothermal curve.

The distinguishing feature of metastability is that, eventually, either via an external perturbation or via a spontaneous fluctuation, a nucleus of the new phase appears, starting an irreversible process which leads to the stable equilibrium state Y, where the phase transition has taken place.

Summary Markov processes and the important subclass of Feller processes are introduced and shown to be determined by the associated semigroups. We take an analytic diversion into semigroup theory and investigate the important concepts of generator and resolvent. Returning to Lévy processes, we obtain two key representations for the generator: first, as a pseudo-differential operator; second, in ‘Lévy–Khintchine form’, which is the sum of a second-order elliptic differential operator and a (compensated) integral of difference operators. We also study the subordination of such semigroups and their action in Lp-spaces.

The structure of Lévy generators, but with variable coefficients, extends to a general class of Feller processes, via Courrège's theorems, and also to Hunt processes associated with symmetric Dirichlet forms, where the Lévy–Khintchine-type structure is apparent within the Beurling–Deny formula.

Markov processes, evolutions and semigroups

Markov processes and transition functions

Intuitively, a stochastic process is Markovian (or, a Markov process) if using the whole past history of the process to predict its future behaviour is no more effective than a prediction based only on a knowledge of the present. This translates into precise mathematics as follows.

Let (Ω, F, P) be a probability space equipped with a filtration (Ft, t ≥ 0). Let X = (X(t), t ≥ 0) be an adapted process.

Summary Section 1.1 is a review of basic measure and probability theory. In Section 1.2 we meet the key concepts of the infinite divisibility of random variables and of probability distributions, which underly the whole subject. Important examples are the Gaussian, Poisson and stable distributions. The celebrated Lévy–Khintchine formula classifies the set of all infinitely divisible probability distributions by means of a canonical form for the characteristic function. Lévy processes are introduced in Section 1.3. These are essentially stochastic processes with stationary and independent increments. Each random variable within the process is infinitely divisible, and hence its distribution is determined by the Lévy–Khintchine formula. Important examples are Brownian motion, Poisson and compound Poisson processes, stable processes and subordinators. Section 1.4 clarifies the relationship between Lévy processes, infinite divisibility and weakly continuous convolution semigroups of probability measures. Finally, in Section 1.5, we briefly survey recurrence and transience, Wiener–Hopf factorisation and local times for Lévy processes.

Review of measure and probability

The aim of this section is to give a brief resumé of key notions of measure theory and probability that will be used extensively throughout the book and to fix some notation and terminology once and for all. I emphasise that reading this section is no substitute for a systematic study of the fundamentals from books such as Billingsley, Itô, Ash and Doléans-Dade, Rosenthal, Dudley or, for measure theory without probability, Cohn.

Summary After a review of first-order differential equations and their associated flows, we investigate stochastic differential equations (SDEs) driven by Brownian motion and an independent Poisson random measure. We establish the existence and uniqueness of solutions under the standard Lipschitz and growth conditions, using the Picard iteration technique. We then turn our attention to investigating properties of the solution. These are exhibited as stochastic flows and as multiplicative cocycles. The interlacing structure is established, and we prove the continuity of solutions as a function of their initial conditions. We then show that solutions of SDEs are Feller processes and compute their generators. Perturbations are studied via the Feynman–Kac formula. We briefly survey weak solutions and associated martingale problems. Finally, we study solutions of Marcus canonical equations and discuss the respective conditions under which these yield stochastic flows of homeomorphisms and diffeomorphisms.

One of the most important applications of Itô's stochastic integral is in the construction of stochastic differential equations (SDEs). These are important for a number of reasons.

1. Their solutions form an important class of Markov processes where the infinitesimal generator of the corresponding semigroup can be constructed explicitly. Important subclasses that can be studied in this way include diffusion and jump-diffusion processes.

2. Their solutions give rise to stochastic flows, and hence to interesting examples of random dynamical systems.

3. They have many important applications to, for example, filtering, control, finance and physics.

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.