The second part of our book is devoted to martingale theory; it is longer than the first and what remains to be written, to fulfil the
promise in our title, is even longer. At any rate the reader will find
here plenty of probability and a (somewhat pale) appearance of the word
potential.
This volume contains the classical martingale theory, for
discrete and continuous time, decomposition theory for supermartingales
and several subjects not mentioned in the first edition: local
martingales, quasimartingales, semimartingales, the spaces W1 and BMO and
Burkholder's inequalities. We also complete the account of general
process theory begun in Chapter IV. We have also included the theory of
stochastic integrals, after some hesitation, for this takes us some way
from potential theory.