shall use the results from the earlier chapters in studying these
processes. In particular, the variation of a process is defined in
terms of its paths. We have the following definitions.
Definition 4.1.2.
a) Let X be a raw process. We call X a process of finite
variation if, for each w, the path X (w): R2 R is a function of
finite variation in the sense of Definition 2.3.1. We define, for X a
process of finite variation, a real-valued process lXI, called the
variation of X by the following:
2
for w e 0, z = (s,t) ER ,
XIZ[(w) = IX.(w)l(s,t)
= IX(o,0)(w) + Var[0s] X.(w)I(,0) + Var[ot]X.(w) (0,.)
SVar[(0,0), (s,t)] *IX.(w) ).
b) If the random variable XIL lim Xi(s,t) 5 + (which
t+*
exists since jXI is increasing in the order sense) is P-integrable, we
say X has integrable variation.
In this chapter, we shall concern ourselves with processes of
integrable variation. We will consider them extended by zero outside
2
R as we did for functions earlier.
Remark. In the book by Dellacherie and Meyer [5], processes of finite
variation are defined as differences of increasing processes. In two
parameters, it seems we might have a problem with this, as we have two