strongly measurable, with integrable variation (in fact, IBx.il
IBI xi by Prop. 4.1.3). By Theorem 4.2.1, E(f 2M d(Bxi)) exists.
R i
n n
Then E E( 21 d(Bx.)) = E E( 2( Mx )dBu) (by Prop. 4.1.3)
i=1 R i i=1 R 1i
n n
= E( E iM x.dB ) = E(f( E 1M xi)dB) = E(fndB ) exists. We proved
i=1 1 u M 1 u u u
in (1) that IXn(w)dB (w) fX (w)dB (w) P-a.s. Moreover, for each n,
ifX'(w)dBu(w), f5IX(w) ldIBI (w) fjx (w)IdIBl (w). By assumption,
the latter is P-integrable, so by Lebesgue JXu(w)dBu(w) is P-
integrable, and we have E(fXn(w)dB (w)) E(JX (w)dB (w)). We have
from before that m(Xn) m(X). It remains to prove that
m(Xn) = E(fx"dB ) for all n. For all z E Z, we have =
k
= * = Z = Em (M.) =
i=1 i 1 xz 1
EE(fM d) = EE() = EE() =
E() = E() = E() =
. (Note: We can now do this last step since
u u
X (w)dB (w) is P-integrable; it was not in part (1)!) Both m(X")
and E(JIXdB ) are Z'-valued, so this means that m(Xn) -
u u
E(f X dB) for all n. Passing to the limit, we obtain
m(X) = E(2 XudBu); in particular, the double integral on the right is
defined.
*