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


 <m(X),z> = E(<J 2X dB ,z>)
 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 +"