stopping time to two-parameter processes yields what is known as a
stopping point.
Definition 1.4.1. A stopping point is a mapping Z: n R2 such that
2
for all v E R, the set Jz < v} belongs to F .
We often denote Z = (S,T). The components S,T are, It turns out,
stopping times with respect to (F ), (F ), respectively, but this by
s t
itself is insufficient to guarantee that (S,T) Is a stopping point.
We do, however, have the following characterization [12]:
Theorem 1.4.2. Let S be an (F )-stopping time, T an (F2)-stopping
5 t
2
time. Then Z = (S,T) is a stopping point if and only if S is F-
measurable, T F -measurable.
Although stopping points have found some application recently
(see, for example Walsh [18] and Fouque [9]), they are rather
inadequate for the purpose of developing a theory of localization for
two-parameter processes. To begin with, due to the partial order
in R2, we are not even assured that jZ>v} F Also, if U and V are
stopping points, UV may not be. Moreover, in one dimension, the graph
of a stopping time T divides R+ x into two components, namely the
stochastic intervals [[O,T)) and [[T,-)), whereas the graph of a
stopping point Z does no such thing. Furthermore, an important
realization of a stopping time is as the debut of a progressive set,
and we have no analogous result in the plane.