74.3. THE MAIN ESTIMATE 2507

+2E

(sup

tm∈Pk

∣∣∣∣∫ tm

0

⟨BX l

k ,dM⟩∣∣∣∣)+∥B∥E ([M] (T ))

Now using the Burkholder Davis Gundy inequality and the inequality for the quadraticvariation of that funny integral involving

⟨BX l

k ,dM⟩,

≤ E (⟨BX0,X0⟩)+E (|e(k)|)+2∥Y∥K′ ∥Xrk∥K

+CE

((∫ T

0

∥∥∥BX lk

∥∥∥2d [M]

)1/2)+∥B∥E ([M] (T ))

Now ∥Bv∥2W ′ ≤ ∥B∥⟨Bv,v⟩. Hence the above reduces to the following after adjusting the

constant C,

≤ E (⟨BX0,X0⟩)+E (|e(k)|)+2∥Y∥K′ ∥Xrk∥K

+CE

((∫ T

0

⟨BX l

k ,Xlk

⟩d [M]

)1/2)+∥B∥E ([M] (T ))

≤ 12

suptm∈Pk

⟨BX (tm) ,X (tm)⟩+(C+∥B∥)E ([M] (T ))

+C (E (⟨BX0,X0⟩) ,∥Y∥K′ ,∥Xrk∥K)+E (|e(k)|)

It follows on subtracting the first term on the right and adjusting constants again,

E

(sup

tm∈Pk

⟨BX (tm) ,X (tm)⟩)

≤ (C+∥B∥)E ([M] (T ))+C (E (⟨BX0,X0⟩) ,∥Y∥K′ ,∥Xrk∥K)+E (|e(k)|)

Now let q→ ∞ and use the monotone convergence theorem which yields the above forun-modified X .

Observe that these partitions are nested and that the constant C (· · ·) is continuous andincreasing in each argument with C (0) = 0,C (· · ·) not depending on T . Thus the left sideis increasing and for given ε > 0, there exists N such that k≥ N implies the right side is nolarger than

C (E (⟨BX0,X0⟩) ,∥Y∥K′ ,∥X∥K)+(C+∥B∥)E ([M] (T ))+ ε (74.3.13)

Now let D denote the union of these nested partitions. Then from the monotone conver-gence theorem,

E(

supt∈D⟨BX (t) ,X (t)⟩

)is no larger than the right side of 74.3.13. Since this is true for all ε > 0, it follows

E(

supt∈D⟨BX (t) ,X (t)⟩

)≤C (E (⟨BX0,X0⟩) ,∥Y∥K′ ,∥X∥K ,E ([M] (T ))) (74.3.14)