2522 CHAPTER 74. A MORE ATTRACTIVE VERSION

Does this formula hold for all t ∈ [0,T ]? Maybe not. However, it will hold for t /∈ Nω . Solet t /∈ Nω .

|⟨BX (t (k)) ,X (t (k))⟩−⟨BX (t) ,X (t)⟩|

≤ |⟨BX (t (k)) ,X (t (k))⟩−⟨BX (t) ,X (t (k))⟩|+ |⟨BX (t) ,X (t (k))⟩−⟨BX (t) ,X (t)⟩|

= |⟨B(X (t (k))−X (t)) ,X (t (k))⟩|+ |⟨B(X (t (k))−X (t)) ,X (t)⟩|

Then using the Cauchy Schwarz inequality on each term,

≤ ⟨B(X (t (k))−X (t)) ,X (t (k))−X (t)⟩1/2

·(⟨BX (t (k)) ,X (t (k))⟩1/2 + ⟨BX (t) ,X (t)⟩1/2

)As before, one can use the lower semicontinuity of

t→ ⟨B(X (t (k))−X (t)) ,X (t (k))−X (t)⟩

on NCω along with the boundedness of ⟨BX (t) ,X (t)⟩ also shown earlier off Nω to conclude

|⟨BX (t (k)) ,X (t (k))⟩−⟨BX (t) ,X (t)⟩|≤ C ⟨B(X (t (k))−X (t)) ,X (t (k))−X (t)⟩1/2

≤C lim infm→∞⟨B(X (t (k))−X (t (m))) ,X (t (k))−X (t (m))⟩1/2 < ε

provided k is sufficiently large. Since ε is arbitrary,

limk→∞

⟨BX (t (k)) ,X (t (k))⟩= ⟨BX (t) ,X (t)⟩ .

It follows that the formula 74.6.34 is valid for all t ∈ NCω . Now define ⟨BX ,X⟩(t) as

⟨BX ,X⟩(t)≡{

⟨B(X (t)) ,X (t)⟩ , t /∈ Nω

The right side of 74.6.34 if t ∈ Nω

Then in short, ⟨BX ,X⟩(t) equals the right side of 74.6.34 for all t ∈ [0,T ] and is conse-quently progressively measurable and continuous. Furthermore, for a.e. t, this functionequals ⟨B(X (t)) ,X (t)⟩. Since it is known on a dense subset, it must be unique.

This implies that t → BX (t) is continuous with values in W ′ for t ∈ NCω . Here is why.

The fact that the formula 74.6.34 holds for all t ∈ NCω implies that t → ⟨BX (t) ,X (t)⟩ is

continuous on NCω . Then for x ∈W, t,s ∈ NC

ω

|⟨BX (t)−BX (s) ,x⟩| ≤ ⟨B(X (t)−X (s)) ,X (t)−X (s)⟩1/2 ∥B∥1/2 ∥x∥W . (74.6.35)

Also

⟨B(X (t)−X (s)) ,X (t)−X (s)⟩= ⟨BX (t) ,X (t)⟩+ ⟨BX (s) ,X (s)⟩−2⟨BX (t) ,X (s)⟩