74.6. THE ITO FORMULA 2523

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

and so the inequality 74.6.35 implies the continuity of t→ BX (t) into W ′ for t /∈ Nω . Notethat by assumption this function is continuous into V ′ for all t.

Now consider the claim about the expectation. Use the function ⟨BX ,X⟩ to define astopping 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]2⟨Y (s) ,X (s)⟩ds+

[R−1BM,M

]τ p(t)+2

∫ t

0X[0,τ p] ⟨BX ,dM⟩ (74.6.36)

The term at the end is now a martingale because X[0,τ p]BX is bounded. Hence the expec-tation of the martingale at the end equals 0. Thus you obtain

E(⟨BX ,X⟩τ p (t)

)= E (⟨BX0,X0⟩)

+E(∫ t

0X[0,τ p]2⟨Y (s) ,X (s)⟩ds

)+E

([R−1BM,M

]τ p(t))

Now use the monotone convergence theorem and the dominated convergence theorem topass to a limit as p→∞ and obtain 74.6.30. The claim about the quadratic variation followsfrom Theorem 66.0.22.

What of the special case where W = H = H ′ and you are in the context of a Gelfandtriple

V ⊆ H = H ′ ⊆V ′

and B is simply the identity. Then we obtain the following theorem as a special case.

Theorem 74.6.3 In Situation 74.1.1 in which W = H = H ′ and B = I, it follows that offa set of measure zero, for every t ∈ [0,T ] , there is a set of measure zero N such that forω /∈ N, there is a continuous function ⟨X ,X⟩ which equals |X (t)|2H for a.e. t such that

⟨X ,X⟩(t) = |X0|2H +∫ t

02⟨Y (s) ,X (s)⟩ds

+[M] (t)+2∫ t

0(X ,dM) (74.6.37)