79.1. THE CASE OF UNIQUENESS 2683

and∫ t

0 ΦndW is continuous and progressively measurable into E hence into V . We can takea subsequence such that ∥Φn−Φ∥L2([0,T ]×Ω;L2(Q1/2U,W)) < 2−n and this will be assumedwhenever convenient.

Note that if Pn is the orthogonal projection onto span( f1, · · · , fn) , then

|PnΦ(y)|W =

∣∣∣∣∣Pn ∑i

∑j

Φi j fi⊗g j (y)

∣∣∣∣∣W

=

∣∣∣∣∣Pn ∑i

∑j

Φi j fi (y,g j)

∣∣∣∣∣W

=

∣∣∣∣∣ n

∑i=1

∑j

Φi j fi (y,g j)

∣∣∣∣∣W

≥

∣∣∣∣∣ n

∑i=1

n

∑j=1

Φi j fi (y,g j)

∣∣∣∣∣W

= |Φn (y)|W

Thus ∣∣∣∣∫ t

sΦndW

∣∣∣∣W≤∣∣∣∣∫ t

sPnΦdW

∣∣∣∣W=

∣∣∣∣Pn

∫ t

sΦdW

∣∣∣∣W≤∣∣∣∣∫ t

sΦdW

∣∣∣∣W.

The following corollary will be useful.

Corollary 79.1.1 Let Φn be as described above. Then

∥Φn (t,ω)∥L2(Q1/2U,W) ≤ ∥Φ(t,ω)∥L2(Q1/2U,W)

where ∥Φn (t,ω)∥L2(Q1/2U,W) ↑ ∥Φ(t,ω)∥L2(Q1/2U,W)

Φ ∈ Lα

(Ω;L∞

([0,T ] ,L2

(Q1/2U,W

)))∩L2

([0,T ]×Ω,L2

(Q1/2U,W

))where α > 2. Then off a set of measure zero, the stochastic integrals

∫ t0 ΦndW satisfy

supn

supt ̸=s

∥∥∫ ts ΦndW

∥∥|t− s|γ

<C (ω) ,γ < 1/2,γ =(α/2)−1

α

Proof: Let, α > 2. As explained above, |∫ r

s ΦndW | ≤ |∫ r

s ΦdW |. Thus by the Burkhol-der Davis Gundy inequality,

supn

∣∣∣∣∫ r

sΦndW

∣∣∣∣≤ ∣∣∣∣∫ r

sΦdW

∣∣∣∣∫

Ω

(∣∣∣∣∫ t

sΦdW

∣∣∣∣)α

dP ≤ C∫

Ω

(∫ t

s∥Φ∥2 dτ

)α/2

dP

≤ C∫

Ω

∥Φ∥α

L∞([0,T ],L2(Q1/2U,H)) |t− s|α/2

≤ C∥Φ∥α

Lα(Ω;L∞([0,T ],L2(Q1/2U,W))) |t− s|α/2

≡ C |t− s|α/2