30.4. MARTINGALES AND SUB-MARTINGALES 825

Hence for ω /∈ N (t) , (t,ω) ∈ A and so Xm (t,ω)→ Y (t,ω) which shows

d (Y (t,ω) ,X (t,ω)) = 0 if ω /∈ N (t) .

The predictable version of X (t) is Y (t). ■Here is a summary of what has been shown above.

adapted and left continuous⇓

predictable⇓

progressively measurable⇓

adapted

Also

stochastically continuous and adapted =⇒ progressively measurable version

30.4 Martingales and Sub-MartingalesThis was done earlier for discreet martingales. The idea here is to consider indiscreet (Whata word to use for a martingale!) ones.

Definition 30.4.1 Let X be a stochastic process defined on an interval I with valuesin a separable Banach space, E. It is called integrable if E (∥X (t)∥) < ∞ for each t ∈ I.Also let Ft be a filtration. An integrable and adapted stochastic process X is called amartingale if for s≤ t

E (X (t) |Fs) = X (s) P a.e. ω.

Recalling the definition of conditional expectation, this says that for F ∈Fs∫F

X (t)dP =∫

FE (X (t) |Fs)dP =

∫F

X (s)dP

for all F ∈ Fs. A real valued stochastic process is called a sub-martingale if whenevers≤ t,

E (X (t) |Fs)≥ X (s) a.e.

and a supermartingale ifE (X (t) |Fs)≤ X (s) a.e.

Example 30.4.2 Let Ft be a filtration and let Z be in L1 (Ω,FT ,P) . Then let X (t) ≡E (Z|Ft).

This works because for s < t, E (X (t) |Fs)≡ E (E (Z|Ft) |Fs) = E (Z|Fs)≡ X (s).

Proposition 30.4.3 The following statements hold for a stochastic process defined on[0,T ]×Ω having values in a real separable Banach space, E.

1. If X (t) is a martingale then ∥X (t)∥ , t ∈ [0,T ] is a sub-martingale.