2166 CHAPTER 63. THE QUADRATIC VARIATION OF A MARTINGALE

Thus on (tnj−1, t

nj ] ξ

n (t) is a bounded martingale. Assuming we are dealing with a rightcontinuous version of this martingale so there are no measurability questions, it followssince A is natural,

E

(∫(tn

j−1,tnj ]

ξn (s)dA(s)

)= E

(∫(tn

j−1,tnj ]

ξn− (s)dA(s)

)

where hereξ

n− (s,ω)≡ lim

r→s−,r∈Dξ

n (s,ω) a.e.

for D≡ ∪∞n=1∪

mnk=1

{tnk

}mnk=0 . Thus, adding these up for all the intervals, (tn

j−1, tnj ] yields

E(∫

(0,a]ξ

n (s)dA(s))= E

(∫(0,a]

ξn− (s)dA(s)

)I want to show that for a.e. ω, ξ

nk (t,ω) converges uniformly to

min(λ ,A(t,ω))≡ λ ∧A(t,ω)

on (0,a]. From this it will follow

E(∫

(0,a]λ ∧A(s,ω)dA(s)

)= E

(∫(0,a]

λ ∧A− (s,ω)dA(s))

Now since s→ A(s,ω) is increasing, there is no problem in writing A− (s,ω) and the aboveequation will suffice to show with simple considerations that for a.e. ω,s→ A(s,ω) is leftcontinuous. Since {A(s)} is a submartingale already, it has a right continuous versionwhich we are using in the above. Thus for a.e. ω it must be the case that s→ A(s,ω) iscontinuous. Let t ∈ (tn

j−1, tnj ]. Then since λ ∧A(t) is Ft measurable,

ξn (t)−λ ∧A(t)≡ E

(λ ∧A

(tn

j)−λ ∧A(t) |Ft

)≥ 0

because A(t) is increasing.Now define a stopping time, T n (ε) for ε > 0 by letting T n (ε) be the infimum of all

t ∈ [0,a] with the property that

ξn (t)−λ ∧A(t)> ε

or if this does not happen, then T n (ε) = a. Thus

T n (ε)(ω) = a∧ inf{t ∈ [0,a] : ξn (t,ω)−λ ∧A(t,ω)> ε}

I need to verify T n (ε) really is a stopping time. Letting s < a, it follows that if ω ∈[T n (ε)≤ s] , then for each N, there exists t ∈ [s,s+ 1

N ) such that ξn (t,ω)−λ ∧A(t,ω)> ε.

Then by right continuity it follows there exists r ∈ D∩ [s,s+ 1N ) such that

ξn (r,ω)−λ ∧A(r,ω)> ε