886 CHAPTER 33. QUADRATIC VARIATION AND STOCHASTIC INTEGRATION

At this point, recall the definition of the covariation. The above equals

14

mn−1

∑k=0

E ( fkgk ([∆Mk +∆Nk]− [∆Mk−∆Nk]))

Rewriting this yields

=14

mn−1

∑k=0

E(

fkgk([(Mτ l )tk+1 +(Nτ l )tk+1 −

((Mτ l )tk +(Nτ l )tk

)]−[(Mτ l )tk+1 − (Nτ l )tk+1 −

((Mτ l )tk − (Nτ l )tk

)]))To save on notation, denote

(Mτ l )tk+1 +(Nτ l )tk+1 −((Mτ l )tk +(Nτ l )tk

)≡ ∆k (Mτ l +Nτ l )

(Mτ l )tk+1 − (Nτ l )tk+1 −((Mτ l )tk − (Nτ l )tk

)≡ ∆k (Mτ l −Nτ l )

Thus the above equals

14

mn−1

∑k=0

E ( fkgk ([∆k (Mτ l +Nτ l )]− [∆k (Mτ l −Nτ l )]))

Now from Corollary 32.4.3,

=14

mn−1

∑k=0

E(

fkgk([∆k (M+N)]τ l − [∆k (M−N)]τ l

))Letting l→ ∞, this reduces to

=14

mn−1

∑k=0

E ( fkgk ([∆k (M+N)]− [∆k (M−N)]))

=14

(∫Ω

∫ t

0f g(d [M+N]−d [M−N])

)=

∫Ω

∫ t

0f gd [M,N]

Now consider the left side of 33.7.

E((∫ t

0f dMτ l ,

∫ t

0gdNτ l

)U

)

≡∫

Ω∑k, j

fkg j ((Mτ l (t ∧ tk+1)−Mτ l (t ∧ tk)) ,(Nτ l(t ∧ t j+1

)−Nτ l (t ∧ t j)

))dP

Then for each ω, the integrand converges as l→ ∞ to

∑k, j

fkg j((M (t ∧ tk+1)−M (t ∧ tk)) ,

(N(t ∧ t j+1

)−N (t ∧ t j)

))