868 CHAPTER 32. QUADRATIC VARIATION
=q
∑k≥p
E((
M (τnk) ,(M(t ∧ τ
nk+1)−M (t ∧ τ
nk)))2)+ mixed terms (32.7)
Consider one of these mixed terms for j < k.
E
M
(τ
nj),
∆ j︷ ︸︸ ︷
M(t ∧ τ
nj+1)−M
(t ∧ τ
nj) ·
M (τnk) ,
∆k︷ ︸︸ ︷
M(t ∧ τ
nk+1)−M (t ∧ τ
nk)
Then it equals
E(E((
M(τ
nj),∆ j)(
M(τ
nj),∆k)|Fτk
))= E
((M(τ
nj),∆ j)
E((
M(τ
nj),∆k)|Fτk
))= E
((M(τ
nj),∆ j)(
M(τ
nj),E(∆k|Fτk
)))= 0
since E(∆k|Fτk
)= E
(M(t ∧ τn
k+1
)−M
(t ∧ τn
k
)|Fτk
)= 0. Now since the mixed terms
equal 0, it follows from 32.7, that expression is dominated by
C2q
∑k≥p
E(∥∥M
(t ∧ τ
nk+1)−M (t ∧ τ
nk)∥∥2)
(32.8)
A mixed term in the above is of the form: For j < k,
E (∆k,∆ j) = E(
E((∆k,∆ j) |Fτn
k
))= E
((∆ j,E
(∆k|Fτn
k
)))= 0
Thus 32.8 equals
C2q
∑k=p
E(∥∥M
(t ∧ τ
nk+1)∥∥2)−E
(∥M (t ∧ τ
nk)∥
2)
= C2E(∥∥M
(t ∧ τ
nq+1)∥∥2−
∥∥M(t ∧ τ
np)∥∥2)
The integrand converges to 0 as p,q→ ∞ and the uniform bound on M allows a use of thedominated convergence theorem. Thus the partial sums of the series of 32.6 converge inL2 (Ω) as claimed.
By adding in the values of{
τn+1k
}Pn (t) can be written in the form
2 ∑k≥0
(M(τ
n+1′k
),(M(t ∧ τ
n+1k+1
)−M
(t ∧ τ
n+1k
)))where τ
n+1′k has some repeats. From the construction,∥∥M
(τ
n+1′k
)−M
(τ
n+1k
)∥∥≤ 2−(n+1)
Thus
Pn (t)−Pn+1 (t) = 2 ∑k≥0
(M(τ
n+1′k
)−M
(τ
n+1k
),(M(t ∧ τ
n+1k+1
)−M
(t ∧ τ
n+1k
)))