process has also been given for square integrable martingales in Meyer

[12], but this too is rather limited.

 Another area that has seen renewed interest is the study of

processes with values in a vector space, especially in a Banach space

(see, for example, Neveu [14]). In Dinculeanu [7] the correspondence

given in Dellacherie and Meyer [5] between processes with finite

variation and stochastic measures is extended to the case where the

processes have values in a Banach space, and in Meyer [12] the

correspondence is stated for two-parameter real-valued processes. In

the one-parameter theory, this result finds its use in applications to

projections, what in turn (at least in the scalar case) is relevant to

the decomposition of supermartingales and to semimartingales. This

correspondence X ix is given by


(1.1.1) p (D) = E(t ddXz)


for  scalar-valued, bounded process. We shall extend this

correspondence to the case where X has integrable variation, with

values in a Banach space E.

 Since the inner integral on the right side of (1.1.1) is computed

pathwise, we begin by studying the properties of Banach-valued
 2
functions f: R E with bounded variation. In Chapter II, we

develop the relevant properties of such functions, and in Chapter III
 2
we show that to each such f we can associate a measure uf: B(R ) E

with finite variation, and we prove the equality


Il( fI' V flr


(1 .1.2)