Remarks.
1) The set A in the definition is not unique: for example, A
and (A,-) always have the same debut.
2) This definition has drawbacks: for example, DA is not
necessarily adherent to A.
3) For an alternate way of defining stopping lines (as a map from
Q into a certain set of curves on R ), see Nualart and Sanz
[15].
4) Since (A,-) is predictable, to say that a stopping line
Z = DA is predictable amounts to saying that [A,m) is
predictable, which is an alternate way of defining predictable
stopping times in one parameter (cf. [4, IV.693).
We introduce a partial order on the set of stopping lines by defining
H S K if (H,m) 3 (K,m).
We also make the convention
H < K if (H,m) [K,w).
The set of stopping lines is then a lattice for 5:
HK is the debut of (H,-) U (K,m), and
HK is the debut of (H,-) l(K,-).
We say that a stopping line H is the limit of an increasing (resp.
decreasing) sequence (H ) of stopping lines if [H,m) = th[H ,=)
n
(resp. if (H,m) = J(Hn,')). In the first case, if we have
n
[H,=) = \'(Hn',), we say that the sequence (H n) announces (foretells)