CHAPTER II
VECTOR-VALUED FUNCTIONS WITH FINITE VARIATION
Since the stochastic integral with respect to a process of finite
(or integrable) variation reduces to taking the Stieltjes integral
pathwise, it is appropriate to study functions defined on R2 (or R2)
with finite variation as a starting point. Throughout this chapter, f
will denote a function defined on R2 with values in a Banach space E,
unless explicitly stated otherwise. We shall write f(z) or f(s,t)
interchangeably for z = (s,t).
2.1 Basic Definitions and Some Examples
For functions of one variable, in order to associate an o-
additive Stieltjes measure, we need the function to be either right or
left continuous. Here we shall use right continuity. In two
dimensions, however, there are two different notions of right
2
continuity: one for the order in R the other a condition merely
sufficient to ensure o-additivity of the associated measure.
Definition 2.1.1. Let f: R2 E be a function.
a) We say that f is right continuous (in the order sense) if, for
all z E R we have
f(z) = lim f(u), or equivalently lim If(u) f(z)- = 0.
u-z u-z
u~z utz