2068 CHAPTER 62. STOCHASTIC PROCESSES

From 62.3.19 it follows from the Borel Cantelli lemma that for fixed t, the set of ω whichare in infinitely many of the sets,[

d (Xm (t) ,X (t))≥ 2−m]has measure zero. Therefore, for each t, there exists a set of measure zero, N (t) such thatfor ω /∈ N (t) and all m large enough

d (Xm (t,ω) ,X (t,ω))< 2−m

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

62.4 MartingalesDefinition 62.4.1 Let X be a stochastic process defined on an interval, I with values in aseparable Banach space, E. It is called integrable if E (||X (t)||)< ∞ for each t ∈ I. Also letFt be a filtration. An integrable and adapted stochastic process X is called a martingale iffor 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 submartingale if whenevers≤ t,

E (X (t) |Fs)≥ X (s) a.e.

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