Proof of Lemma 2
Cheng Ouyang
April 9, 2017
We give the proof of the second lemma that is needed in order to establish the martingale representation theorem.
Lemma 0.1. Let f : Rn → R be a bounded smooth function with bounded derivatives and
X = f (Bs1 , Bs2 − Bs1 , ..., Bsn − Bs1 ). Then there is a progressively measurable process H
such that
Z
1
Hs dBs .
X = EX +
0
Proof. We prove by induction. The case when n = 1 is given by Lemma 1 proved in class.
By induction hypothesis applied to the Brownian motion {Bs+s1 − Bs1 , 0 ≤ s ≤ 1 − s1 },
we have
Z
1
Hsx dBs ,
f (x, Bs2 − Bs1 , ..., Bsn − Bs1 ) = h(x) +
s1
where
h(x) = Ef (x, Bs2 − Bs1 , ..., Bsn − Bs1 ),
and the dependence of H x on x is smooth. Hence, replacing x by Bs1 we have
Z
1
X = h(Bs1 ) +
Hs dBs ,
s1
Bs1
where Hs = Hs
; s1 ≤ s ≤ 1. We also have for some Hs ; 0 ≤ s ≤ s1
Z s1
h(Bs1 ) = Eh(Bs1 ) +
Hs dBs .
0
It is clear that Eh(Bs1 ) = EX, hence
Z
X = EX +
Hs dBs .
0
The proof is thus completed.
1
1