4.4 On the Equality Iml = B
In Theorem 4.2.3, we began with a stochastic function B with
integrable variation, and associated a measure m with finite
variation, and we proved that Iml S IBI'. We now consider some cases
where this is in fact an equality.
Theorem 4.4.1 Let E,F be two Banach spaces and Z C F' a subspace
norming for F. Let B: R xf + L(E,F) be a right continuous stochastic
function satisfying conditions (1) and (ii) of Theorem 4.2.3, and let
m: M L(E,Z') be the corresponding measure with finite variation Iml
satisfying
= E()
R+
for any E-valued measurable process X E LE (|, ). We have the
equality Iml = p ,B i.e.,
Im ( lx) = E(f 2 IdIv v ) for X E LE (ml)
R
in each of the following cases:
1) There is a lifting p of P such that p[B ] = B for
z a
R2
2) E is separable and there is a countable subset S C Z norming
for F.
3) E is separable and B x is integrable for every x E E and
z
z E R 2
) s measurable and B is ntegrable for z R
4) B is measurable and B is integrable for z R2
a +"