4.3 Vector-Valued Stochastic Functions Associated
With Measures
In this section we consider the converse; starting with a
stochastic measure m with finite variation, we will find a stochastic
function B with integrable variation such that m is associated with B
in the sense of Theorem 4.2.3. The precise result is the following:
Theorem 4.3.1. Let E,F be two Banach spaces and ZE F' a subspace
norming for F. Let m: M L(E,F) be a stochastic measure with finite
variation Im|. Then there exists a right continuous stochastic
function B: R x *- L(E,Z') satisfying:
i) B has raw Integrable variation IBI.
ii) For every x c E and z c Z, is a real-valued raw
process with Integrable variation II.
Moreover, we having the following:
1) If X Is an E-valued measurable process we have X E L (m)
E
if and only if X E LE(UIB). In this case the Integral
E() is defined for every z E Z,
= E(< 2X dB ,z>), and
+
Im ( lx) = E(fR 2 I IdBI ), i.e., ml um8 .
2) If F is separable (or more generally if Bx is separably
valued for every x E E), then Bx is measurable for every x E E.
If B is separably valued, then B is measurable.