2174 CHAPTER 63. THE QUADRATIC VARIATION OF A MARTINGALE

Now let o(1) denote any quantity which converges to 0 as ε → 0 for all λ ∈ [−p, p] andO(1) is a quantity which is bounded as ε → 0. Then from 63.8.45 and 63.8.46 you canconsider the power series for eiλy j which converges uniformly due to 63.8.43 and write63.8.46 as

E

(eiλX(τε∧t j−1)

(1− λ

2

2σ

2j (1+o(1))

)).

then noting that from 63.8.44 which shows σ2j is o(1) ,it is routine to verify

1− λ2

2σ

2j (1+o(1)) = e−

λ22 σ2

j (1+o(1)).

Now this shows

E(

eiλX(τε∧t j))= E

(eiλX(τε∧t j−1)e−

λ22 σ2

j (1+o(1)))

Recall that σ2j ≤ δ n = t j− t j−1. Consider∣∣∣∣E (eiλX(τε∧t j)

)−E

(eiλX(τε∧t j−1)e−

λ22 δ n

)∣∣∣∣=

∣∣∣∣E(eiλX(τε∧t j−1)e−λ22 σ2

j (1+o(1)))−E

(eiλX(τε∧t j−1)e−

λ22 δ n

)∣∣∣∣=

∣∣∣∣E(eiλX(τε∧t j−1)(

e−λ22 σ2

j (1+o(1))− e−λ22 δ n

))∣∣∣∣=

∣∣∣∣E(eiλX(τε∧t j−1)e−λ22 δ n

(e−

λ22 σ2

j (1+o(1))+ λ22 δ n −1

))∣∣∣∣≤ E

(∣∣∣∣e λ22 (δ n−σ2

j)+σ2j o(1)−1

∣∣∣∣)Everything in the exponent is o(1) and so the above expression is bounded by

O(1)E

(∣∣∣∣∣λ 2

2(δ n−σ

2j)+σ

2jo(1)

∣∣∣∣∣)

≤ O(1)E((

δ n−σ2j)+δ n |o(1)|

)= O(1)

[δ n−E

(y2

j)+δ no(1)

]. (63.8.47)

Therefore, ∣∣∣∣E (eiλX(τε∧t j))−E

(eiλX(τε∧t j−1)e−

λ22 δ n

)∣∣∣∣≤ O(1)

[δ n−E

(y2

j)+δ no(1)

]