73.7. THE ITO FORMULA 2495

and so the inequality 73.7.40 implies the continuity of t→ BX (t) into W ′ for t /∈ Nω . Notethat by assumption this function is continuous into V ′ for all t. It was also shown that it isweakly continuous into W ′ on [0,T ] and hence it is bounded in W ′.

Now consider the claim about the expectation. Since the stochastic integral

2∫ t

0

(Z ◦ J−1)∗BX ◦ JdW

is only a local martingale, it is necessary to employ a stopping time. We use the function⟨BX ,X⟩ to define this stopping time as

τ p ≡ inf{t > 0 : ⟨BX ,X⟩(t)> p}

This is the first hitting time of a continuous process and so it is a valid stopping time. Usingthis, leads to

⟨BX ,X⟩τ p (t) = ⟨BX0,X0⟩+∫ t

0X[0,τ p] (s)

(2⟨Y (s) ,X (s)⟩+ ⟨BZ,Z⟩L2

ds)

ds

+2∫ t

0X[0,τ p] (s)

(Z ◦ J−1)∗BXτ p ◦ JdW (73.7.41)

By continuity of ⟨BX ,X⟩ ,τ p = ∞ for all p large enough. Take expectation of both sidesof the above. In the integrand of the last term, BX refers to the function BX (t,ω) ≡B(X (t,ω)) and so it is progressively measurable because X is assumed to be so. HenceBXτ p is also progressively measurable and for a.e. Also, for a.e. s,

∥∥BX (s∧ τ p)∥∥

W ′ ≤√p√∥B∥. Therefore, one can take expectations and get

E(⟨BX ,X⟩τ p (t)

)= E (⟨BX0,X0⟩)

+E(∫ t

0X[0,τ p] (s)

(2⟨Y (s) ,X (s)⟩+ ⟨BZ,Z⟩L2

ds)

ds)

Now let p→ ∞ and use the monotone convergence theorem on the left and the dominatedconvergence theorem on the right to obtain the desired result 73.7.35. The claim about thequadratic variation follows from Corollary 65.11.1.