2148 CHAPTER 63. THE QUADRATIC VARIATION OF A MARTINGALE

where the stopping times were defined such that τnk+1 is the first time t > τn

k such that∣∣M (t)−M(τn

k

)∣∣2 = 2−n and τn0 = 0. Recall that limk→∞ τn

k = ∞ or T in the way it wasformulated earlier. Then it was shown that Pn (t) converged to a martingale P(t) in L1 (Ω).Then by the usual procedure using the Borel Cantelli lemma, a subsequence converges toP(t) uniformly off a set of measure zero. It is easy to estimate Pn (t) .

|Pn (t)| ≤ ∑k≥0

∣∣M (t ∧ τnk+1)∣∣2−|M (t ∧ τ

nk)|

2 = |M (t)|2 ≤M∗

This follows from the observation that(M(t ∧ τ

nk+1),M (t ∧ τ

nk))≤ 1

2

(∣∣M (t ∧ τnk+1)∣∣2 + |M (t ∧ τ

nk)|

2)

Then it follows that supt∈[0,T ] |P(t)(ω)| ≤M∗ (ω)≤C for a.e. ω. The quadratic variation[M] was defined as

|M (t)|2 = P(t)+ [M] (t)

Thus [M] (t) ≤ 2(M∗)2. Now consider the above limit in 63.6.26. From the assumptionthat M is uniformly bounded,∫ T

0∥RMn−RM∥2 d [M]≤

∫ T

04C2d [M] = 4C2 [M] (T )≤ 4C2 (2C2)< ∞

Also, by the continuity of the martingale, for each ω,

limn→∞∥RMn−RM∥2 = 0

By the dominated convergence theorem, and the fact that the integrand is bounded,

limn→∞

∫ T

0∥RMn−RM∥2 d [M] = 0.

Then from the above estimate and the dominated convergence theorem again, 63.6.26 fol-lows. Thus RM ∈ GM .

From the above lemma, it makes sense to speak of∫ t

0(RM)dM

and this is a continuous martingale having values in R. Also from the above argument, if{tnk

}mnk=0 is a sequence of partitions such that

limn→∞

max{∣∣tn

i − tni+1∣∣ , i = 0, · · · ,mn

}= 0,

then it follows thatmn−1

∑i=0

RM (ti)(M (t ∧ tk+1)−M (t ∧ tk))→∫ t

0(RM)dM

in L2 (Ω), this for each t ∈ [0,T ].Now here is the main result.