63.3. THE COVARIATION 2129

=14([M+N]τ − [M−N]τ

)=

14([Mτ +Nτ ]− [Mτ −Nτ ])≡ [Mτ ,Nτ ] .

The really interesting part is the next equality. This will involve Corollary 63.2.3.

[M,N]τ − [Mτ ,N] = [Mτ ,Nτ ]− [Mτ ,N]

≡ 14([Mτ +Nτ ]− [Mτ −Nτ ])− 1

4([Mτ +N]− [Mτ −N])

=14([Mτ +Nτ ]+ [Mτ −N])− 1

4([Mτ +N]+ [Mτ −Nτ ]) , (63.3.11)

the difference of two increasing adapted processes. Also, this equals

local martingale − (Mτ ,N)+(Mτ ,Nτ)

Claim: (Mτ ,N)− (Mτ ,Nτ) = (Mτ ,N−Nτ) is a local martingale. Let σn be a localizingsequence for both M and M. Such a localizing sequence is of the form τM

n ∧τNn where these

are localizing sequences for the indicated local submartingale. Then obviously,

(−(Mτ ,N)+(Mτ ,Nτ))σn =−(Mσn∧τ ,Nσn

)+(Mσn∧τ ,Nσn∧τ

)where Nσn and Mσn are martingales. To save notation, denote these by M and N respec-tively. Now use Lemma 63.1.1. Let σ be a stopping time with two values.

E ((Mτ (σ) ,N (σ)−Nτ (σ))) = E (E ((Mτ (σ) ,N (σ)−Nτ (σ)) |Fτ))

Now Mτ (σ) is M (σ ∧ τ) which is Fτ measurable and so by the Doob optional samplingtheorem,

= E (Mτ (σ) ,E (N (σ)−Nτ (σ) |Fτ))

= E (Mτ (σ) ,N (σ ∧ τ)−N (τ ∧σ)) = 0

whileE ((Mτ (t) ,N (t)−Nτ (t))) = E (E ((Mτ (t) ,N (t)−Nτ (t)) |Fτ))

Since Mτ (t) is Fτ measurable,

= E ((Mτ (t) ,E (N (t)−Nτ (t) |Fτ)))

= E ((Mτ (t) ,E (N (t ∧ τ)−N (t ∧ τ)))) = 0

This shows the claim is true.Now from 63.3.11 and Corollary 63.3.3,

[M,N]τ − [Mτ ,N] = 0.

Similarly[M,N]τ − [M,Nτ ] = 0