2520 CHAPTER 74. A MORE ATTRACTIVE VERSION

[R−1BM,M

](t)+2

∫ t

0⟨BX ,dM⟩ (74.6.29)

Also, there exists a unique continuous, progressively measurable function which is denotedhere as ⟨BX ,X⟩ such that it equals ⟨BX (t) ,X (t)⟩ for a.e. t and ⟨BX ,X⟩(t) equals the rightside of the above for all t. In addition to this,

E (⟨BX ,X⟩(t)) =

E (⟨BX0,X0⟩)+E(∫ t

02⟨Y (s) ,X (s)⟩ds+

[R−1BM,M

](t))

(74.6.30)

Also the quadratic variation of the stochastic integral in 74.6.29 is dominated by∫ t

0∥BX∥2

W ′ d [M] (74.6.31)

Also t→ BX (t) is continuous with values in W ′ for t ∈ NCω .

Proof: Let t ∈ NCω \D. For t > 0, let t (k) denote the largest point of Pk which is less

than t. Suppose t (m)< t (k). Hence m≤ k. Then

BX (t (m)) = BX0 +∫ t(m)

0Y (s)ds+BM (t (m)) ,

a similar formula holding for X (t (k)) . Thus for t > t (m) , t ∈ NCω ,

B(X (t)−X (t (m))) =∫ t

t(m)Y (s)ds+B(M (t)−M (t (m)))

which is the same sort of thing studied so far except that it starts at t (m) rather than at 0and BX0 = 0. Therefore, from Lemma 74.6.1 it follows

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

=∫ t(k)

t(m)2⟨Y (s) ,X (s)−X (t (m))⟩ds

+[R−1BM,M

](t (k))−

[R−1BM,M

](t (m))

+2∫ t(k)

t(m)⟨B(X−X (t (m))) ,dM⟩ (74.6.32)

Consider that last term. It equals

2∫ t(k)

t(m)

⟨B(

X−X lm

),dM

⟩(74.6.33)

This is dominated by

2∣∣∣∣∫ t(k)

0

⟨B(

X−X lm

),dM

⟩−∫ t(m)

0

⟨B(

X−X lm

),dM

⟩∣∣∣∣