2482 CHAPTER 73. THE HARD ITO FORMULA, IMPLICIT CASE

Then from the assumption that τ p = ∞ for all p large enough, it follows that

A = ∪∞m=1A∩ ([τm = ∞]\ [τm−1 < ∞])

Now

P(A∩ [τm = ∞])≤ P(

ω :∫ T

0X[0,τm]

∣∣∣(Z ◦ J−1)∗BX ◦ J∣∣∣2 dt = ∞

)(73.6.21)

Consider the integrand. What is the meaning of∣∣(Z ◦ J−1

)∗BX ◦ J∣∣2? You have

(Z ◦ J−1)∗ ∈L2

(W ′,J

(Q1/2U

)′)while BX ∈W ′ and so

(Z ◦ J−1

)∗BX ∈L2

(J(Q1/2U

)′,R)

which is just(J(Q1/2U

))′.

Thus(Z ◦ J−1

)∗BX ◦ J would be in(Q1/2U

)′and to get the L2 norm, you would take an

orthonormal basis in Q1/2U denoted as {gi} and the square of this norm is just

∑i

[((Z ◦ J−1)∗BX ◦ J

)(gi)]2≡ ∑

i

[(Z ◦ J−1)∗BX (Jgi)

]2

≡ ∑i

[BX(Z ◦ J−1 (Jgi)

)]2= ∑

i[(BX)(Zgi)]

2

≤ ∑i∥BX∥2 ∥Zgi∥2

W

Now incorporating the stopping time, you know that for a.e. t,

⟨BX ,X⟩(t) = ⟨BX (t) ,X (t)⟩ ≤ m

and so ∥BX (t)∥can be estimated in terms of m as follows.

|⟨B(X (t)) ,w⟩| ≤ ⟨B(X (t)) ,X (t)⟩1/2 ∥B∥1/2 ∥w∥W

=

(∑

i⟨BX (t) ,ei⟩2V ′,V

)1/2

∥B∥1/2 ∥w∥W

≤√

m∥B∥1/2 ∥w∥W , so ∥BX (t)∥ ≤ m∥B∥1/2

Thus the integrand satisfies for a.e. t

X[0,τm]

∣∣∣(Z ◦ J−1)∗BX ◦ J∣∣∣2 ≤ m∥B∥∥Z∥2

L2

Hence, from 73.6.21, P(A∩ [τm = ∞])

≤ P(

ω :∫ T

0∥Z∥2

L2m∥B∥dt = ∞

)