2476 CHAPTER 73. THE HARD ITO FORMULA, IMPLICIT CASE

≤ C+12

E

(sup

tm∈Pk

⟨B(

X lk

)τ p(tm) ,

(X l

k

)τ p(tm)

⟩)+C∥Z∥2

L 2([0,T ]×Ω,L2)+E (|e(k)|) .

It follows that

12

E

(sup

tm∈Pk

⟨B(

X lk

)τ p(tm) ,

(X l

k

)τ p(tm)

⟩)≤C+E (|e(k)|)

Now let p→ ∞ and use the monotone convergence theorem to obtain

E

(sup

tm∈Pk

⟨BX l

k (tm) ,Xlk (tm)

⟩)= E

(sup

tm∈Pk

⟨BX (tm) ,X (tm)⟩)≤C+E (|e(k)|)

(73.4.14)As mentioned above, this constant C is a continuous function of

||Y ||K′ , ||X ||K , ||Z||J ,∥⟨BX0,X0⟩∥L1(Ω,H)

and equals zero when all of these quantities equal 0 and is increasing with respect to eachof the above quantities. Also, for each ε > 0,

E

(sup

tm∈Pk

⟨BX (tm) ,X (tm)⟩)≤C+ ε

whenever k is large enough.Let D denote the union of all the Pk. Thus D is a dense subset of [0,T ] and it has just

been shown, since the Pk are nested, that for a constant C dependent only on the abovequantities which is independent of Pk,

E(

supt∈D⟨BX (t) ,X (t)⟩

)≤C+ ε.

Since ε > 0 is arbitrary,

E(

supt∈D⟨BX (t) ,X (t)⟩

)≤C (73.4.15)

Thus, enlarging N, for ω /∈ N,

supt∈D⟨BX (t) ,X (t)⟩=C (ω)< ∞ (73.4.16)

where∫

ΩC (ω)dP < ∞. By Lemma 69.4.1, there exists a countable set {ei} of vectors in

V such that ⟨Bei,e j

⟩= δ i j

and for each x ∈W,

⟨Bx,x⟩=∞

∑i=0⟨Bx,ei⟩2 , Bx =

∞

∑i=1⟨Bx,ei⟩Bei