CHAPTER 1
INTRODUCTION
Families of random variables indexed by directed sets have been
objects of study for some time. The most common by far, though, has
2 2
been R and especially the first quadrant of the plane, as this
is considered the most "natural" extension of the usual indexing set R
or R One of the principal objects of study relating to such
processes has been the stochastic integral of processes indexed by the
plane.
Stochastic integration with respect to two-parameter Brownian
motion has been studied extensively; the focus of more recent study
has been the more general problem of extending the one-parameter
theory of stochastic integration with respect to a semimartingale. In
one parameter, this is done by writing a semimartingale X as
X = M + A
with M a locally square integrable local martingale and A a process of
finite variation (see, for example, Dellacherie and Meyer [5]). The
major problem in two parameters is with the notion of "local martingale":
the theory of stopping is not sufficiently well-developed to permit a
definition with all the necessary properties. Some preliminary work
has been done, however, using the notion of an increasing path (see,
for example, Fouque [9] and Walsh [18]). A definition of a stopped