2448 CHAPTER 72. THE HARD ITO FORMULA
−m−1
∑j=1
∣∣X (t j+1)−X (t j)−
(M(t j+1
)−M (t j)
)∣∣2 (72.4.8)
where e(k)→ 0 for a.e. ω and also in L1 (Ω).Now it follows on discarding the negative terms,
supt j∈Pk
∣∣X (t j)∣∣2 ≤ |X0|2 +2
∫ T
0|⟨Y (u) ,X r
k (u)⟩|du
+2 supt∈[0,T ]
∣∣∣∣∫ t
0R((
Z (u)◦ J−1)∗X lk (u)
)◦ JdW
∣∣∣∣+mk−1
∑j=0
∣∣∣∣∫ t j+1
t j
Z (u)dW∣∣∣∣2
where there are mk +1 points in Pk.Do
∫Ω
to both sides. Using the Ito isometry, this yields
∫Ω
(sup
t j∈Pk
∣∣X (t j)∣∣2)dP ≤ E
(|X0|2
)+2 ||Y ||K′ ||X
rk ||K
+mk−1
∑j=0
∫ t j+1
t j
∫Ω
||Z (u)||2 dPdu
+2∫
Ω
(sup
t∈[0,T ]
∣∣∣∣∫ T
0R((
Z (u)◦ J−1)∗X lk (u)
)◦ JdW
∣∣∣∣)
dP+E (|e(k)|)
≤C+∫ T
0
∫Ω
||Z (u)||2 dPdu+
+2∫
Ω
(sup
t∈[0,T ]
∣∣∣∣∫ T
0R((
Z (u)◦ J−1)∗X lk (u)
)◦ JdW
∣∣∣∣)
dP
≤C+2∫
Ω
(sup
t∈[0,T ]
∣∣∣∣∫ T
0R((
Z (u)◦ J−1)∗X lk (u)
)◦ JdW
∣∣∣∣)
dP
where the result of Lemma 72.2.1 that X rk converges to X̄ in K shows that the term which
is the product 2 ||Y ||K′∣∣∣∣X r
k
∣∣∣∣K is bounded. The constant C is a continuous function of
||Y ||K′ , ||X̄ ||K , ||Z||J ,∥X0∥L2(Ω,H)
which equals zero when all are equal to zero. The term involving the stochastic integral isnext.
Applying the Burkholder Davis Gundy inequality, Theorem 63.4.4 for F (r) = r alongwith the description of the quadratic variation of the Ito integral found in Corollary 65.11.1∫
Ω
supt j∈Pk
|X (tk)|2 dP
≤ C+C∫
Ω
(∫ T
0
∣∣∣∣∣∣R((Z (u)◦ J−1)∗X lk (u)
)◦ J∣∣∣∣∣∣2 du
)1/2
dP