It is four years since the first version of this book appeared and there has continued to be intense activity focused on Lévy processes and related areas. One way of gauging this is to look at the number of books and monographs which have appeared in this time. Regarding fluctuation theory of Lévy processes, there is a new volume by A. Kyprianou and the St Flour lectures of R. Doney. From the point of view of interactions with analysis, N. Jacob has published the third and final volume of his impressive trilogy. Applications to finance has continued to be a highly active and fast moving area and there are two new books here – a highly comprehensive and thorough guide by R. Cont and P. Tankov and a helpful introduction aimed at practioners from W. Schoutens. There have also been new editions of classic texts by Jacod and Shiryaev and Protter.

Changes to the present volume are of two types. On the one hand there was the need to correct errors and typos and also to make improvements where this was appropriate. In this respect, I am extremely grateful to all those readers who contacted me with remarks and suggestions. In particular I would like to thank Fangjun Xu, who is currently a first-year graduate student at Nanzai University, who worked through the whole book with great zeal and provided me with an extremely helpful list of typos and mistakes.

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–Itô 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. We then obtain necessary and sufficient conditions for a Lévy process to be of finite variation and also to have finite moments. 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 Appendix 2.9 at the end of the chapter.

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. The existence of Lyapunov exponents for solutions of SDES will be investigated.

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

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

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.

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, local times for Lévy processes, regular variation and subexponentiality.

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 Doléns-Dade, Rosenthal, Dudley or, for measure theory without probability, Cohn.

Summary We will now study the stochastic integration of predictable processes against martingale-valued measures. Important examples are the Brownian, Poisson and Lévy-type cases. In the case where the integrand is a sure function, we investigate the associated Wiener–Lévy integrals, particularly the important example of the Ornstein–Uhlenbeck process and its relationship with self-decomposable random variables. In Section 4.4, we establish Itô's formula, which is one of the most important results in this book. Immediate spin-offs from this are Lévy's characterisation of Brownian motion, Burkholder's inequality and estimates for stochastic integrals. We also introduce the Stratonovitch, Marcus and backwards stochastic integrals and indicate the role of local time in extending Itô's formula beyond the class of twice-differentiable functions.

Integrators and integrands

In Section 2.6, we identified the need to develop a theory of integration against martingales that is not based on the usual Stieltjes integral. Given that our aim is to study stochastic differential equations driven by Lévy processes, our experience with Poisson integrals suggests that it might be profitable to integrate against a class of real-valued independently scattered martingale-valued measures M defined on (S, I). Here S = ℝ+ × E, where E ∈ B(ℝd) and I is the ring comprising finite unions of sets of the form I × A where A ∈ B(E) and I is itself a finite union of intervals.

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 and this is then applied to obtain the chaos decomposition for multiple Wiener–Lévy integrals. We then give a brief introduction to Malliavin calculus in the Brownian case. The final section of this chapter 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.

Abstract

Introduction

Muller's ratchet is a mechanism that has been suggested as an explanation for the evolution of sex [13]. The idea is simple; in an asexually reproducing population chromosomes are passed down as indivisible blocks and so the number of deleterious mutations accumulated along any ancestral line in the population can only increase.