2132 CHAPTER 63. THE QUADRATIC VARIATION OF A MARTINGALE

whereτ ≡ inf{t ∈ [0,T ] : ||M (t)||> λ}

Thus N is a martingale and N (0)= 0. In fact N (t)= 0 as long as t ≤ τ . As usual inf( /0)≡∞.Note

[τ < ∞] = [M∗ > λ ]⊇ [N∗ > 0] .

This is because to say τ < ∞ is to say there exists t < T such that ||M (t)||> λ which is thesame as saying M∗ > λ . Thus the first two sets are equal. If τ = ∞, then from the formulafor N (t) above, N (t) = 0 for all t ∈ [0,T ] and so it can’t happen that N∗ > 0. Thus the thirdset is contained in [τ < ∞] as claimed.

Let β > 2 and let δ ∈ (0,1) . Then

β −1 > 1 > δ > 0

Consider the following which is set 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 63.3.3,

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 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 62.11.3 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 ]) .