2112 CHAPTER 62. STOCHASTIC PROCESSES

Therefore, one can extract a subsequence{

Mnk

}such that

P

(sup

t∈[0,T ]

∣∣∣∣Mnk (t)−Mnk+1 (t)∣∣∣∣> 2−k

)≤ 2−k.

By the Borel Cantelli lemma, it follows{

Mnk (t)(ω)}

converges uniformly on [0,T ] fora.e. ω . Denote by M (t)(ω) the thing to which it converges, a continuous process becauseof the uniform convergence. Also, because it is the pointwise limit off a set of measurezero, ω →M (t)(ω) is Ft measurable. Also, from 62.12.49 and Fatou’s lemma∫

Ω

supt∈[0,T ]

||Mn (t)−M (t)||p dP

≤ lim infk→∞

∫Ω

supt∈[0,T ]

∣∣∣∣Mn (t)−Mnk (t)∣∣∣∣p dP≤ ε

whenever n is large enough, this from the assumption that {Mn } is Cauchy. Thus

limn→∞

E

(sup

t∈[0,T ]||Mn (t)−M (t)||p

)= 0

and so for each t,Mn (t)→M (t) in Lp (Ω). This also shows that for large, n

E

(sup

t∈[0,T ]||M (t)||p

)≤ E

(sup

t∈[0,T ](||M (t)−Mn (t)||+ ||Mn (t)||)p

)

≤ 2p−1E

(sup

t∈[0,T ]||M (t)−Mn (t)||p + sup

t∈[0,T ](||Mn (t)||)p

)< ∞

It only remains to verify M is a martingale. Let s≤ t and let B ∈Fs. For each s, Mn (s)→M (s) in Lp (Ω). Then from the above, ω →M (s)(ω) is Fs measurable. Then it followsthat ∫

BM (s)dP = lim

n→∞

∫B

Mn (s)dP = limn→∞

∫B

E (Mn (t) |Fs)dP

= limn→∞

∫B

Mn (t)dP =∫

BM (t)dP

and so by definition, E (M (t) |Fs) = M (s) which shows M is a martingale.

Proposition 62.12.3 The functions M (t) for each M ∈M pT (E) are equi integrable.

Proof: This follows because∫[||M(t)||≥λ ]

||M (t)||p dP≤∫[supt∈[0,T ]||M(t)||≥λ ]

(sup

t∈[0,T ]||M (t)||p

)dP (62.12.50)