CHAPTER IV
VECTOR-VALUED PROCESSES
WITH FINITE VARIATION
An important part of the general theory of processes in one
parameter is the correspondence between processes of finite variation
and measures on R+xn (see for example Dellacherie and Meyer [5, VI.
64-89] and also Kussmaul [10]). This correspondence finds
applications in the notion of dual projections of processes, which are
used in the theory of potentials and in decomposition of
supermartingales (see for example Dellacherle and Meyer [5, nos. VI.
71-113], also Rao [17] and Metivier [11]).
In the one-parameter case, the extension of the correspondence to
Banach-valued processes is done in Dellacherie and Meyer [5]. In two
parameters, the correspondence for real-valued processes is stated
(more or less) in Meyer [12]; we shall presently give a more directly
applicable (for our purposes) version, along with a proof, as the case
of finite variation on R2 is more delicate (as we have seen). In
fact, many times, in the literature results are given for increasing
processes, and then extended by defining a process of finite variation
as a difference of two increasing processes. The method we use here
is a little more constructive.