72.4. THE MAIN ESTIMATE 2451

+2∫ tm

0R((

Z (u)◦ J−1)∗X lk (u)

)◦ JdW +

m−1

∑j=0

∣∣M (t j+1)−M (t j)

∣∣2−

m−1

∑j=1

∣∣X (t j+1)−X (t j)−

(M(t j+1

)−M (t j)

)∣∣2 (72.4.10)

where e(k)→ 0 in L1 (Ω) and a.e. ω .Can you obtain something similar even in case X0 is not assumed to be in Lp (Ω,V )?

Let Z0k ∈ Lp (Ω,V )∩L2 (Ω,H) ,Z0k→ X0 in L2 (Ω,H) . Then

|X (t1)−X0| ≤ |X (t1)−Z0k|+ |Z0k−X0|

Also, restoring the superscript to identify the parition,

X(

tk1

)−Z0k = X0−Z0k +

∫ tk1

0Y (s)ds+

∫ tk1

0Z (s)dW.

Of course ∥X̄−Z0k∥K is not bounded but for each k it is at least finite. There is a sequenceof partitions Pk,∥Pk∥ → 0 such that all the above holds. In the definitions of K,K′,Jreplace [0,T ] with [0, t] and let the resulting spaces be denoted by Kt ,K′t ,Jt . Let nk denotea subsequence of {k} such that

∥X̄−Z0k∥Ktnk1

< 1/k.

Then from the above lemma,

E

 supt∈[0,t

nk1 ]

∣∣X (tnk1

)−Z0k

∣∣2H

≤ C

(||Y ||K′

tnk1

,∥X̄−Z0k∥Ktnk1

, ||Z||Jtnk1

,∥X0−Z0k∥L2(Ω,H)

)

≤ C

(||Y ||K′

tnk1

,1k, ||Z||J

tnk1

,∥X0−Z0k∥L2(Ω,H)

)

HenceE(∣∣X (tnk

1

)−X0

∣∣2)≤ 2E(∣∣X (tnk

1

)−Z0k

∣∣2H

)+2E

(|Z0k−X0|2H

)≤ 2C

(||Y ||K′

tnk1

,1k, ||Z||J

tnk1

,∥X0−Z0k∥L2(Ω,H)

)+2∥Z0k−X0∥2

which converges to 0 as k→ ∞. It follows that there exists a suitable subsequence suchthat 72.4.10 holds even in the case that X0 is only known to be in L2 (Ω,H). From now on,assume this subsequence for the paritions Pk. Thus k will really be nk.