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.