31.5. CONTINUOUS SUB-MARTINGALE CONVERGENCE 851

Theorem 31.5.3 Let {X (t)} be a continuous sub-martingale adapted to a normalfiltration such that

supt{E (|X (t)|)}=C < ∞.

Then there exists X∞ ∈ L1 (Ω) such that

limt→∞

X (t)(ω) = X∞ (ω) a.e. ω.

Proof: Let U[a,b] be defined by

U[a,b] = limM→∞

UM[a,b].

Thus the random variable U[a,b] is an upper bound for the number of upcrossings. FromLemma 31.5.2 and the assumption of this theorem, there exists a constant C independentof M such that

E(

UM[a,b]

)≤ C

b−a+1.

Letting M→ ∞, it follows from monotone convergence theorem that

E(U[a,b]

)≤ C

b−a+1

also. Therefore, there exists a set of measure 0 Nab such that if ω /∈Nab, then U[a,b] (ω)<∞.That is, there are only finitely many upcrossings. Now let

N = ∪{Nab : a,b ∈Q} .

It follows that for ω /∈ N, it cannot happen that

lim supt→∞

X (t)(ω)− lim inft→∞

X (t)(ω)> 0

because if this expression is positive, there would be arbitrarily large values of t whereX (t)(ω) > b and arbitrarily large values of t where X (t)(ω) < a where a,b are rationalnumbers chosen such that

lim supt→∞

X (t)(ω)> b > a > lim inft→∞

X (t)(ω)

Thus there would be infinitely many upcrossings which is not allowed for ω /∈N. Therefore,the limit limt→∞ X (t)(ω) exists for a.e. ω . Let X∞ (ω) equal this limit for ω /∈ N and letX∞ (ω) = 0 for ω ∈ N. Then X∞ is measurable and by Fatou’s lemma,∫

Ω

|X∞ (ω)|dP≤ lim infn→∞

∫Ω

|X (n)(ω)|dP <C. ■

Now here is an interesting result of Doob.

Theorem 31.5.4 Let {M (t)} be a continuous real martingale adapted to the nor-mal filtration Ft . Then the following are equivalent.

1. The random variables M (t) are equi-integrable.