2480 CHAPTER 73. THE HARD ITO FORMULA, IMPLICIT CASE

Also, restoring the superscript to identify the parition,

B(

X(

tk1

)−Z0k

)= B(X0−Z0k)+

∫ tk1

0Y (s)ds+B

∫ tk1

0Z (s)dW.

Of course ∥X−Z0k∥K is not bounded, but for each k it is finite. There is a sequence ofpartitions Pk,∥Pk∥ → 0 such that all the above holds. In the definitions of K,K′,J re-place [0,T ] with [0, t] and let the resulting spaces be denoted by Kt ,K′t ,Jt . Let nk denote asubsequence of {k} such that

∥X−Z0k∥Ktnk1

< 1/k.

Then from the above lemma,

E(⟨

B(X(tnk1

)−Z0k

),X(tnk1

)−Z0k

⟩)≤C

(||Y ||K′

tnk1

,∥X−Z0k∥Ktnk1

, ||Z||Jtnk1

,⟨B(X0−Z0k) ,X0−Z0k⟩L1(Ω)

)

≤C

(||Y ||K′

tnk1

,1k, ||Z||J

tnk1

,⟨B(X0−Z0k) ,X0−Z0k⟩L1(Ω)

)Hence

E(⟨

B(X(tnk1

)−X0

),X(tnk1

)−X0

⟩)≤ 2E

(⟨B(X(tnk1

)−Z0k

),X(tnk1

)−Z0k

⟩)+2E (⟨B(Z0k−X0) ,Z0k−X0⟩)

≤ 2C

(||Y ||K′

tnk1

,1k, ||Z||J

tnk1

,⟨B(X0−Z0k) ,X0−Z0k⟩L1(Ω)

)+2∥B∥∥Z0k−X0∥2

L2(Ω,W )

which converges to 0 as k→ ∞. It follows that there exists a suitable subsequence suchthat 73.5.18 holds even in the case that X0 is only known to be in L2 (Ω,W ). From now on,assume this subsequence for the partitions Pk. Thus k will really be nk and it suffices toconsider the limit as k→ ∞ of the equation of 73.5.18. To emphasize this point again, thereason for the above observations is to argue that, even when X0 is only in L2 (Ω,W ) , onecan neglect

⟨B(X (t1)−X0−M (t1)) ,X (t1)−X0−M (t1)⟩

in passing to the limit as k→ ∞ provided a suitable subsequence is used.

73.6 ConvergenceThe question is whether the above stochastic integral

∫ t0(Z ◦ J−1

)∗BX ln ◦JdW converges as

n→ ∞ in some sense to ∫ t

0

(Z ◦ J−1)∗BX ◦ JdW (73.6.19)