2156 CHAPTER 63. THE QUADRATIC VARIATION OF A MARTINGALE

Lemma 63.7.7 Let A be an increasing adapted stochastic process which is right continu-ous. Also let ξ (t) be a bounded right continuous martingale. Then

E (ξ (t)A(t)) = E(∫

(0,t]ξ (s)dA(s)

)

and A is natural, if and only if for all such bounded right continuous martingales,

E (ξ (t)A(t)) = E(∫

(0,t]ξ (s)dA(s)

)= E

(∫(0,t]

ξ− (s)dA(s))

Lemma 63.7.8 Let (Ω,F ,P) be a probability space and let G be a σ algebra contained inF . Suppose also that { fn} is a sequence in L1 (Ω) which converges weakly to f in L1 (Ω) .That is, for every h ∈ L∞ (Ω) ,

∫Ω

fnhdP→∫

Ω

f hdP.

Then E ( fn|G ) converges weakly in L1 (Ω) to E ( f |G ).

Proof:First note that if h ∈ L∞ (Ω,F ) , then E (h|G ) ∈ L∞ (Ω,G ) because if A ∈ G ,

∫A|E (h|G )|dP≤

∫A

E (|h| |G )dP =∫

A|h|dP

and so if A = [|E (h|G )|> ||h||∞] , then if P(A)> 0,

||h||∞

P(A)<∫

A|E (h|G )|dP≤

∫A|h|dP≤ ||h||

∞P(A) ,

a contradiction. Hence P(A)= 0 and so E (h|G )∈L∞ (Ω,G ) as claimed. Let h∈L∞ (Ω,G ) .

∫Ω

E ( fn|G )hdP =∫

Ω

E (E ( fn|G )h|G )dP

=∫

Ω

E ( fn|G )E (h|G )dP

=∫

Ω

E ( fnE (h|G ) |G )dP

=∫

Ω

fnE (h|G )dP