33.6. THE CASE OF f ∈ L2 (Ω× [0,T ] ;L2 (U,H)) 905

It is not necessary to have f be uniformly bounded. Instead, one can consider f ∈L2 (Ω× [0,T ] ;L2 (U,H)) where f is progressively measurable. I considered the casewhere f is continuous in t above because it is a convenient way to tie this in to the or-dinary theory of Stieltjes integrals and to point out that these standard objects do delivermartingales in some reasonable cases.

As before, I will first consider the case where M is a bounded martingale and then ex-tend to the case where M is a local martingale. As before, it is all based on the fundamentalinequality

12

E

(sup

t∈[0,T ]

∥∥∥∥∫ t

0f dM

∥∥∥∥2)≤ E

(∥∥∥∥∫ T

0f dM

∥∥∥∥2)≤∫

Ω

∫ T

0∥ f∥2

L2dtdP (33.31)

which was shown to hold for all adapted f continuous in t and also bounded, which impliesthe integral on the right is finite.

Let f ∈ L2 (Ω× [0,T ] ;L2 (U,H)) be adapted and let f (t,ω) be extended as 0 for t off[0,T ].

Definition 33.6.1 Let fn (t)≡ n∫ t

t−1/n f (s)ds.

Lemma 33.6.2 fn is adapted and t→ fn (t) is continuous for a.e. ω .

Proof: First consider the claim about continuity. For each ω off a set of measure zero,f ∈ L2 ([0,T ] ;L2) and so, for such ω

∥ fn (t)− fn (t̂)∥2L2

=

∥∥∥∥n∫ t

t−1/nf (s)ds−n

∫ t̂

t̂−1/nf (s)ds

∥∥∥∥2

dP

= n∥∥∥∥∫ t̂−1/n

t−1/nf (s)ds+

∫ t̂

tf (s)ds

∥∥∥∥2

dP

≤ 2n

(∥∥∥∥∫ t̂−1/n

t−1/nf (s)ds

∥∥∥∥2

+

∥∥∥∥∫ t̂

tf (s)ds

∥∥∥∥2)≤ 8n(t− t̂)∥ f∥2

L2([0,T ];L2)

It follows that ω → n∫ t

t−1/n f (s)ds is Ft measurable because X[0,t] f is Ft ×B ([0,T ])measurable by assumption that f is progressively measurable. ■

Observe that

∥ fn− f∥L2(Ω×[0,T ];L2(U,H)) ≡(∫ T

0

∫Ω

∥ fn− f∥2L2

dPdt)1/2

=

(∫Ω

∫ T

0

∥∥∥∥n∫ 0

−1/n( f (t + s)− f (t))ds

∥∥∥∥2

L2

dtdP

)1/2

From Minkowski’s inequality,

≤ n∫ 0

−1/n

(∫Ω

∫ T

0∥ f (t + s)− f (t)∥2 dtdP

)1/2

ds (33.32)

so

∥ fn− f∥2L2(Ω×[0,T ];L2(U,H)) ≤

∫Ω

n∫ 0

−1/n

∫ T

0∥ f (t + s)− f (t)∥2 dtdsdP