63.4. THE BURKHOLDER DAVIS GUNDY INEQUALITY 2133

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 ] .

Then there exists t < T such that [M] (t)> β2λ

2. This time, let

N (t)≡ [M] (t)− [Mτ ] (t)−||M (t)−Mτ (t)||2

This is still a martingale since by Corollary 63.3.3

[M] (t)− [Mτ ] (t) = [M−Mτ ] (t)

Claim: N (t)(ω) hits λ2(

1−δ2)

for some t < T for ω ∈ Sr.

Proof of claim: Fix such a ω ∈ Sr. Let t < T be such that [M] (t)> β2λ

2. Then t > τ

and so for that ω,

N (t) > β2λ

2−λ2−||M (t)−M (τ)||2

≥ (β −1)2λ

2− (||M (t)||+ ||M (τ)||)2

≥ (β −1)2λ

2− r2δ

2λ

2 ≥ λ2−δ

2λ

2

By the intermediate value theorem, it hits λ2(

1−δ2). This proves the claim.

Claim: N (t)(ω) never hits −δ2λ

2 for ω ∈ Sr.Proof of claim: By Corollary 63.3.3, if it did at t, then t > τ because N (t) = 0 for

t ≤ τ, and so

0 ≤ [M] (t)− [Mτ ] (t) = ||M (t)−M (τ)||2−δ2λ

2

≤ (||M (t)||+ ||M (τ)||)2−δ2λ

2 ≤ r2δ

2λ

2−δ2λ

2 < 0,

a contradiction. This proves the claim.It follows that for each r ∈ (0,1) ,

P(Sr)≤ P(

N (t) hits λ2(

1−δ2)

before −δ2λ

2)