7.3 Strategies, Semi-martingales and Stochastic Integration 127(i) H·Vis a local martingale and
(ii)H·Nis a local martingale.
The second one is standard. The jumps ofYandH·Yare bounded by 1
and hence for a predictable stopping timeTwe have ∆BT=E[∆YT|FT−]
is bounded by 1. Also|HT∆YT|≤1 and henceE[HT∆YT|FT−]=HT∆BT
is bounded by 1. ForT totallyinaccessiblewehave∆BT = 0 a.s. as B
is predictable and henceHT∆NT =HT∆YT as well as ∆NT =∆YTare
bounded by 1. It follows that|H∆N|and|∆N|are bounded by 2. The local
martingaleN is locallyL^2 and the increasing process
∫t
0 H′
ud[N, N]uHuis
also locallyL^2 .TheL^2 theory of martingales shows that (H·N)isalocal
martingale (even locallyL^2 ).
The first part is more tricky. For eachpwe defineRp=inf{t|
∫t
0 |HudBu|≥
p}. This makes sense sinceH·Bexists as an ordinary integral. The sequence
(Rp)∞p=1increases to infinity. AsH·Bhas jumps bounded by 1 we have
∫Rp
0 |HudBu|≤p+1.
We also defineSp=inf{t|
∫t
0 |HudUu|≥p}. BecauseH·Uexists as an
ordinary integral this makes sense again and also (Sp)∞p=1increases to infinity.
For eachnwe now putHn=H (^1) {|H|≤n}. Clearly
∫Rp
0 |H
n
udBu|≤p+1 and
because of the hypothesis (H,∆M)≥θwe have
(
(Hn·U)Sp)−
≤p+|θ|.We
now take one more sequence (τp)∞p=1of stopping times increasing to infinity
so thatVτp ∈H^1 (P). BecauseHnis bounded, (Hn·V)τpis anH^1 (P)-
martingale and henceE
[
(Hn·V)τp∧Sp∧Rp]
=0forallnandp. However,
for the negative part we find (Hn·V)−τp∧Sp∧Rp≤(Hn·B)−τp∧Sp∧Rp+(Hn·
U)−τp∧Sp∧Rp≤p+1+p+|θ|=2p+1+|θ|. ThereforeE
[
(Hn·V)−τp∧Sp∧Rp]
≤
2 p+1+E[|θ|] and the same holds for (Hn·V)+τp∧Sp∧Rp. This in turn implies
E
[∣
∣(Hn·V)τp∧Sp∧Rp∣
∣
]
≤ 4 p+2+2E[|θ|]. Ifn→∞,wehave((Hn−H)·V)∗=supt|((Hn−H)·V)t|. This implies that (Hn·V)τp∧Sp∧Rp
P
→0 and an application
of Fatou’s lemma yields that alsoE
[∣
∣(H·V)τp∧Sp∧Rp∣
∣
]
≤ 4 p+2+2E[|θ|].
Now the definition ofSpandRpimply that|(H·V)t|≤ 2 pfort<τp∧Sp∧Rp.
We finally deduce thatE
[
sup 0 ≤t≤τp∧Sp∧Rp|(H·V)t|]
≤ 2 p+2(2(p+1)+2 E[|θ|])<∞.
The proof is almost complete now. The processW =Vτp∧Sp∧Rpis an
H^1 -martingale, the integral (H·W) has a maximal functionf=(H·W)∗
that is integrable. This is sufficient to prove thatH·W is a martingale and
hence anH^1 -martingale. Again we use approximations. For eachnletνnbe
defined asνn=inf{t||(Hn·W)t−(H·W)t|> 1 }. It is clear thatνn→∞.
Each (Hn·W)νn is a martingale and its maximal function is bounded by
(Hn·W)∗νn≤(H·W)∗νn+1+2f. The last term coming from a possible jump
atνn. It follows that for allnwe have the inequality (Hn·W)∗νn≤ 3 f+1.
A simple application of Lebesgue’s dominated convergence theorem allows us
to conclude thatH·Wis a martingale. The proof is now complete.