2494 CHAPTER 73. THE HARD ITO FORMULA, IMPLICIT CASE

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 73.7.39 is valid for all t /∈ Nω . Now define ⟨BX ,X⟩(t) as

⟨BX ,X⟩(t)≡{

⟨B(X (t)) ,X (t)⟩ , t /∈ Nω

The right side of 73.7.39 if t ∈ Nω

Then in short, ⟨BX ,X⟩(t) equals the right side of 73.7.39 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 /∈ Nω . Here is why.The fact that the formula 73.7.39 holds for all t /∈ Nω implies that t → ⟨BX (t) ,X (t)⟩ iscontinuous on NC

ω . Then for x ∈W,

|⟨BX (t)−BX (s) ,x⟩| ≤ ⟨B(X (t)−X (s)) ,X (t)−X (s)⟩1/2 ∥B∥1/2 ∥x∥W . (73.7.40)

Also

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

By weak continuity of t→ BX (t) shown earlier,

limt→s⟨BX (t) ,X (s)⟩= ⟨BX (s) ,X (s)⟩ .

Therefore,limt→s⟨B(X (t)−X (s)) ,X (t)−X (s)⟩= 0