32.3. THE QUADRATIC VARIATION 867

time and denote Mα p∧δ by M to save notation. Thus M will be uniformly bounded andfrom the definition of the stopping times τn

k , for t ∈ [0,T ] ,

M (t)≡ ∑k≥0

M(t ∧ τ

nk+1)−M (t ∧ τ

nk) , (32.4)

and the terms of the series are eventually 0, as soon as τnk = ∞.

Therefore,

∥M (t)∥2 =

∥∥∥∥∥∑k≥0

M(t ∧ τ

nk+1)−M (t ∧ τ

nk)

∥∥∥∥∥2

Then this equals= ∑

k≥0

∥∥M(t ∧ τ

nk+1)−M (t ∧ τ

nk)∥∥2

+ ∑j ̸=k

((M(t ∧ τ

nk+1)−M (t ∧ τ

nk)),(M(t ∧ τ

nj+1)−M

(t ∧ τ

nj)))

(32.5)

Consider the second sum. It equals

2 ∑k≥0

k−1

∑j=0

((M(t ∧ τ

nk+1)−M (t ∧ τ

nk)),(M(t ∧ τ

nj+1)−M

(t ∧ τ

nj)))

= 2 ∑k≥0

(M (t ∧ τnk+1)−M (t ∧ τ

nk)),

telescopesk−1

∑j=0

(M(t ∧ τ

nj+1)−M

(t ∧ τ

nj))

= 2 ∑k≥0

((M(t ∧ τ

nk+1)−M (t ∧ τ

nk)),M (t ∧ τ

nk))

This last sum equals Pn (t) defined as

2 ∑k≥0

(M (τn

k) ,(M(t ∧ τ

nk+1)−M (t ∧ τ

nk)))≡ Pn (t) (32.6)

This is because in the kth term, if t ≥ τnk , then it reduces to(

M (τnk) ,(M(t ∧ τ

nk+1)−M (t ∧ τ

nk)))

while if t < τnk , then the term reduces to ((M (t)−M (t)) ,M (t)) = 0 which is also the same

as (M (τn

k) ,(M(t ∧ τ

nk+1)−M (t ∧ τ

nk)))

.

This is a finite sum because eventually, for large enough k, τnk = T . However the number

of nonzero terms depends on ω . This is not a good thing. However, a little more can besaid. In fact the sum in 32.6 converges in L2 (Ω). Say ∥M (t,ω)∥ ≤C.

E

( q

∑k≥p

(M (τn

k) ,(M(t ∧ τ

nk+1)−M (t ∧ τ

nk))))2

