Two of the most fundamental concepts in the theory of stochastic processes are the Markov property and the martingale property* This book is written
for readers who are acquainted with both of these ideas in the discrete-time
setting, and who now wish to explore stochastic processes in their continuous-
time context. It has been our goal to write a systematic and thorough
exposition of this subject, leading in many instances to the frontiers of knowledge.
At the same time, we have endeavored to keep the mathematical prerequisites
as low as possible, namely, knowledge of measure-theoretic probability and
some familiarity with discrete-time processes. The vehicle we have chosen for
this task is Brownian motion, which we present as the canonical example of
both a Markov process and a martingale. We support this point of view by
showing how, by means of stochastic integration and random time change,
all continuous-path martingales and a multitude of continuous-path Markov
processes can be represented in terms of Brownian motion. This approach
forces us to leave aside those processes which do not have continuous paths.
Thus, the Poisson process is not a primary object of study, although it is
developed in Chapter 1 to be used as a tool when we later study passage times
and local time of Brownian motion.