2168 CHAPTER 63. THE QUADRATIC VARIATION OF A MARTINGALE

Thus it suffices to take the supremum over the half open interval, [0,a). It follows

[Qn > ε] = [T n (ε)< a]

By right continuity,ξ

n (T n (ε))−λ ∧A(T n (ε))≥ ε

on [Qn > ε] .

εP([Qn > ε]) = εP([T n (ε)< a])

≤∫[Qn>ε]

(ξ n (T n (ε))−λ ∧A(T n (ε)))dP

≤∫

Ω

(ξ n (T n (ε))−λ ∧A(T n (ε)))dP

Therefore, from 63.7.40,

P([Qn > ε]) ≤ 1ε

∫Ω

(λ ∧A(⌈T n (ε)⌉)−λ ∧A(T n (ε)))dP

≤ 1ε

∫Ω

(A(⌈T n (ε)⌉)−A(T n (ε)))dP (63.7.41)

By optional sampling theorem,

E (M (T n (ε))) = E (M (0)) = 0

and alsoE (M (⌈T n (ε)⌉)) = E (M (0)) = 0.

Therefore, 63.7.41 reduces to

P([Qn > ε])≤ 1ε

∫Ω

(X (⌈T n (ε)⌉)−X (T n (ε)))dP

By the assumption that {X (t)} is DL, it follows the functions in the above integrand areequi integrable and so since limn→∞ X (⌈T n (ε)⌉)−X (T n (ε)) = 0, the above integral con-verges to 0 as n→ ∞ by Vitali’s convergence theorem, Theorem 11.5.3 on Page 257. Itfollows that there is a subsequence, nk such that

P([

Qnk > 2−k])≤ 2−k

and so from the definition of Qn,

limk→∞

supt∈[0,a]

|ξ nk (t)−λ ∧A(t)|

giving uniform convergence. Now recall that

E(∫

(0,a]ξ

nk (s)dA(s))= E

(∫(0,a]

ξnk− (s)dA(s)

)