Our discrete models are only a crude approximation to the way in which stock markets actually move. A better model would be one in which stock prices can change at any instant. As early as 1900 Bachelier, in his thesis ‘La théorie de la spéculation’, proposed Brownian motion as a model of the fluctuations of stock prices. Even today it is the building block from which we construct the basic reference model for a continuous time market. Before we can proceed further we must leave finance to define and construct Brownian motion.
Our first approach will be to continue the heuristic of §2.6 by considering Brownian motion as an ‘infinitesimal’ random walk in which smaller and smaller steps are taken at ever more frequent time intervals. This will lead us to a natural definition of the process. A formal construction, due to Lévy, will be given in §3.2, but this can safely be omitted. Next, §3.3 establishes some facts about the process that we shall require in later chapters. This material too can be skipped over and referred back to when it is used.
Just as discrete parameter martingales play a key rôle in the study of random walks, so for Brownian motion we shall use continuous time martingale theory to simplify a number of calculations; §3.4 extends our definitions and basic results on discrete parameter martingales to the continuous time setting.