63.7. DOOB MEYER DECOMPOSITION 2157

and so

limn→∞

∫Ω

E ( fn|G )hdP = limn→∞

∫Ω

fnE (h|G )dP

=∫

Ω

f E (h|G )dP

=∫

Ω

E ( f E (h|G ) |G )dP

=∫

Ω

E (h|G )E ( f |G )dP

=∫

Ω

E (E ( f |G )h|G )dP

=∫

Ω

E ( f |G )hdP

and this proves the lemma.Next suppose {X (t)} is a real submartingale and suppose X (t) = M (t)+A(t) where

A(t) is an increasing stochastic process adapted to Ft such that A(0) = 0 and {M (t)} is amartingale adapted to Ft . Also let T be a stopping time bounded above by a. Then by theoptional sampling theorem, and the observation that {|M (t)|} is a submartingale∫

[|X(T )|≥λ ]|X (T )|dP

≤∫[|X(T )|≥λ ]

|M (T )|dP+∫[|X(T )|≥λ ]

A(T )dP

≤∫[|X(T )|≥λ ]

E (|M (a)| |FT )dP+∫[|X(T )|≥λ ]

E (A(a) |FT )dP

≤∫[|X(T )|≥λ ]

|M (a)|dP+∫[|X(T )|≥λ ]

A(a)dP

Now by Theorem 60.6.4,

P([|XT | ≥ λ ])≤ 2λ

E (|X (0)|+ |X (a)|)

and so P([|X (T )| ≥ λ ])→ 0 uniformly for T a stopping time bounded by a as λ → ∞ andso this shows equi integrability of {X (T )} because A(t,ω)≥ 0.

This motivates the following definition.

Definition 63.7.9 A stochastic process, {X (t)} is called DL if for all a > 0, the set ofrandom variables, {X (T )} for T a stopping time bounded by a is equi integrable.

Example 63.7.10 Let {M (t)} be a continuous martingale. Then {M (t)} is of class DL.

To show this, let a > 0 be given and let T be a stopping time bounded by a. Then bythe optional sampling theorem, M (0) ,M (T ) ,M (a) is a martingale and so

E (M (a) |FT ) = M (T )