1.4.2 Stopping Lines
Let AC R 2x be a random set (the usual definition: 1A is a
measurable process). The open envelope of A, denoted (A,-), is the
random set whose section for each w c 0 is given by
(A,-)(w) = ) (z,-).
zeA(w)
Some properties of (A,-) (proofs can be found in Meyer [12]):
1) If A is progressive, (A,t) is predictable.
2) The interior of a progressive set A is progressive, from which
we obtain, by passing to complements, that the closure of A is
progressive.
We designate in particular by [A,=) the closure of (A,-) (i.e., for
each w, we define [A,")(w) = (A,w)(w)). [A,-) is progressive If A is
progressive, by the above properties; it is called the closed envelope
of A. The random set
DA [A,-)\ (A,-)
is called the debut of A: it is progressive if A is progressive.
Definition 1.4.3. a) A random set Z is called a stopping line If it
is the debut of a progressive set, i.e., if there is a progressive set
A such that
Z = DA = [A,-)\ (A,m).
b) A stopping line Z is predictable if it belongs (as a set) to the
predictable o-field P.