(a) Welcome back on board! You will have noticed that for this second leg of your journey, there are two pilots rather than one. D.W. is sure that you will be as delighted as he is that control is being shared with L.C.G.R.—amongst so many other things, just the man for a Wiley excursion!

We apologize for the considerable delay in departure. Anyone who knows what has been happening to British universities will need no further explanation, and will share our sadness.

(b) The book is meant to help the research student reach the stage where he or she can begin both to think up and tackle new problems and to read the up-to-date literature across a wide spectrum; and to persuade him or her that it is worth the effort.

We can say that we ourselves find the subject sufficiently good fun to have enjoyed the task of writing. (We even had some amusement from typing the manuscript ourselves with the very basic non-mathematical word-processor VIEW on the BBC micro. Occasionally, we got into trouble when trying to use global editing to substitute the most commonly occurring phrases for shorthand versions of our own devising. But, in the main, we were very satjto's formulaied!)

(c) Chapter IV, Introduction to Itô calculus, is particularly concerned with developing the theory of the stochastic integral (of a previsible process) with respect to a continuous semimartingale, and with giving a large number of applications.

Here, we give the gist of the ‘martingale and stochastic integral’ method, and illustrate its use via a large number of fully-worked examples. We do not apologize for sometimes advertising the method by showing how it can obtain results which are well known and elementary. Thus, for example, we take the trouble to prove some standard results about the humble Markov chain with finite state-space. But we have also tried to bring into this chapter applications which are less elementary, and which hint at the excitement of the subject today.

TERMINOLOGY AND CONVENTIONS

R-processes and L-processes

We now use the term R-process on [0, ∞) to signify a process all of whose paths are right-continuous on [0, ∞) with limits from the left on (0, ∞). Thus an R-process is what was called in Volume 1 a Skorokhod process, and what is called elsewhere a càdlàg process, or a corlol process, or whatever. An R-function or R-path on [0, ∞) is defined via the obvious analogous definition.

The L-processes on (0, ∞), all of whose paths are left-continuous with limits from the right, will now begin to feature largely in the theory.

This chapter is a reminder of what every probabilist should know, with the emphasis on things that tend to be neglected. The considerable length of the chapter—and it is now much more extensive than in the first edition—should be sufficient guarantee that ‘reminder’ is used in the usual ‘courtesy’ sense! Because things are now developed in strictly logical order, you may sometimes have to wait a little time for applications. (We very occasionally cheat just a little in Exercises by using things that you may feel are not yet proved with full rigour, but we always clear these up later.) Exercises are very much part of the text—please do them!

So many reminders of standard definitions are included that we break with our usual definition format except when we wish to give special emphasis to particularly important material that may not be so familiar.

BASIC MEASURE THEORY

The basic results of measure theory are summarised here, with commentary, but mostly without proofs. A full account, with all results proved, may be found, for example, in Williams [15], referred to as [W] throughout this chapter; that account has the advantage that its notation and terminology are the same as those used here. Neveu [1] is a marvellous account of measure theory for probabilists; and, for the definitive account of the full theory, including Choquet capacitability theory (which is needed for the Debut and Section Theorems), see Volume 1 of Dellacherie and Meyer [1].

Long ago (or so it seems today), Chung wrote on page 196 of his book [1]: ‘One wonders if the present theory of stochastic processes is not still too difficult for applications.’ Advances in the theory since that time have been phenomenal, but these have been accompanied by an increase in the technical difficulty of the subject so bewildering as to give a quaint charm to Chung's use of the word ‘still’. Meyer writes in the preface to his definitive account of stochastic integral theory: ‘ …il faut … un cours de six mois sur les définitions. Que peut on y faire?’

I have thought up as intuitive a picture of the subject as I can, written it down at speed, and refused to be lured back by piety (or even by wit!) to cancel half a line. ‘First’ intuition, which is what you need when you are learning the subject, is raw, rough and ready; and, as you have guessed, I make the excuse that it demands a compatible style and lack of polish.

Note that I wrote ‘first intuition’. Consider an example. Meyer's concept of a right process is exactly right for Markov process theory, but the concept is the result of a long evolution. To understand it properly, you need a highly developed intuition, and that takes time to acquire. The difficulty with the best advanced literature is that its authors have too much intuition; never make the mistake of thinking otherwise.

1. What is Brownian motion, and why study it? The first thing is to define Brownian motion. We assume given some probability triple (Ω, F, P).

(1.1) DEFINITION. A real-valued stochastic process {Bt,:t∈R+} is a Brownian motion if it has the properties

(1.2) (i) B0(ω) = 0, ∀ω

(1.2) (ii) the map t ↦ Bt(ω) is a continuous function of t∈R+ for all ω

(1.2)(iii) for every t,h ≥ 0, Bt+h, – Bt, is independent of {Bu: 0 ≤ u ≤ t}, and has a Gaussian distribution with mean 0 and variance h.

The conditions (1.2)(ii) and (1.2)(iii) are the really essential ones; if B = {Bt: t∈R+} is a Brownian motion, we frequently speak of {ξ + Bt,:t∈R+} as a Brownian motion (started at ξ); the starting point ξ, can be a fixed real, or a random variable independent of B.

Now that we know what a Brownian motion is, questions of existence and uniqueness (answered in Section 6) are less important than an answer to the second question of the title, ‘Why study it?’ There are many answers to this question, but to us there seem to be four main ones:

1. (i) Virtually every interesting class of processes contains Brownian motion—Brownian motion is a martingale, a Gaussian process, a Markov process, a diffusion, a Lévy process,…;

2. (ii) Brownian motion is sufficiently concrete that one can do explicit calculations, which are impossible for more general objects;

3. (iii) Brownian motion can be used as a building block for other processes (indeed, a number of the most important results on Brownian motion state that the most general process in a certain class can be obtained from Brownian motion by some sequence of transformations);

4. (iv) last but not least, Brownian motion is a rich and beautiful mathematical object in its own right.