876 CHAPTER 32. QUADRATIC VARIATION

Let β > 2 and let δ ∈ (0,1) . Then β −1 > 1 > δ > 0. Consider the following which isset up to use the good lambda inequality.

Sr ≡ [M∗ > βλ ]∩[([M] (T ))1/2 ≤ rδλ

]where 0 < r < 1.It is shown that Sr corresponds to hitting “this before that” and there is anestimate for this which involves P([N∗ > 0]) which is bounded above by P([M∗ > λ ]) asdiscussed above. This will satisfy the hypotheses of the good lambda inequality.

Claim: For ω ∈ Sr, N (t) hits λ2(

1−δ2).

Proof of claim: For ω ∈ Sr, there exists a t < T such that ∥M (t)∥ > βλ and so usingCorollary 32.4.3 and triangle inequality,

N (t) ≥ |∥M (t)∥−∥Mτ (t)∥|2− [M−Mτ ] (t)≥ |βλ −λ |2− [M] (t)

≥ (β −1)2λ

2−δ2λ

2

which shows that N (t) hits (β −1)2λ

2 − δ2λ

2 for ω ∈ Sr. By the intermediate valuetheorem, it also hits λ

2(

1−δ2)

. This proves the claim.

Claim: N (t)(ω) never hits −δ2λ

2 for ω ∈ Sr.

Proof of claim: Suppose t is the first time N (t) reaches −δ2λ

2. Then t > τ becauseN (t) = 0 on [0,τ] and so

N (t) = −δ2λ

2 ≥ |∥M (t)∥−λ |2− [M] (t)+ [Mτ ] (t)

≥ −r2λ

2δ

2,

a contradiction since r < 1. This proves the claim.Therefore, for all ω ∈ Sr, N (t)(ω) reaches λ

2(

1−δ2)

before it reaches −δ2λ

2. Itfollows

P(Sr)≤ P(

N (t) reaches λ2(

1−δ2)

before −δ2λ

2)

and because of Theorem 31.6.3 this is no larger than

P([N∗ > 0])δ

2λ

2

λ2(

1−δ2)−(−δ

2λ

2) = P([N∗ > 0])δ

2 ≤ δ2P([M∗ > λ ]) .

ThusP([M∗ > βλ ]∩

[([M] (T ))1/2 ≤ rδλ

])≤ P([M∗ > λ ])δ

2

By the good lambda inequality,∫Ω

F (M∗)dP≤C∫

Ω

F(([M] (T ))1/2

)dP

which is one half the inequality.Now consider the other half. This time define the stopping time τ by

τ ≡ inf{

t ∈ [0,T ] : ([M] (t))1/2 > λ

}and let

Sr ≡[([M] (T ))1/2 > βλ

]∩ [2M∗ ≤ rδλ ] .