(i.e., the variation of the measure associated with f is equal to the
measure associated with the variation of f). In Chapter IV we prove
that a stochastic measure can be associated with a process of
integrable variation, in the same manner as in Dinculeanu [7], and we
consider the converse problem: that of associating a function with a
given stochastic measure. Unfortunately, the equality (1.1.2) is not
preserved in general for processes and stochastic measures, so we end
up by establishing some sufficient conditions for the equality to hold.
1.1 Notation
2
The index set is R ; we shall sometimes consider functions and
processes extended by zero to all of R In the rare cases where we
look at points outside the first quadrant, we shall say so
explicitly. We denote points in R2 by z, u, w, v and their
coordinates by the letters r, s, t, p. For example, then, we write
z = (s,t), u (p,r), etc.
There are two notations commonly used in the literature for the
order relation in R2: we adopt here the notation of Meyer [12]. For
two points z = (s,t), z' = (s',t'), we have z.z' iff sSs', tSt';
z