2106 CHAPTER 62. STOCHASTIC PROCESSES

and tn→∞, Egoroff’s theorem implies that there exists a set E of measure less than δ suchthat on EC, the convergence of the M (tn) is uniform. Thus∫

Ω

|M (tm)−M (tn)|dP =∫

E|M (tm)−M (tn)|dP+

∫EC|M (tm)−M (tn)|dP

≤ 2ε

5+∫

EC|M (tm)−M (tn)|dP < ε

whenever m,n are large enough. Therefore, the sequence {M (tn)} is Cauchy in L1 (Ω)which implies it converges to something in L1 (Ω) which must equal M (∞) a.e.

Next suppose there is a function M (∞) to which M (t) converges in L1 (Ω) . Then for tfixed and A ∈Ft , then as s→ ∞,s > t∫

AM (t)dP =

∫A

E (M (s) |Ft)dP≡∫

AM (s)dP

→∫

AM (∞)dP =

∫A

E (M (∞) |Ft)

which shows E (M (∞) |Ft) = M (t) a.e. since A ∈Ft is arbitrary. By Lemma 62.7.11,∫[|M(t)|≥λ ]

|M (t)|dP =∫[|M(t)|≥λ ]

|E (M (∞) |Ft)|dP

≤∫[|M(t)|≥λ ]

E (|M (∞)| |Ft)dP

=∫[|M(t)|≥λ ]

|M (∞)|dP (62.10.37)

Now from this,

λP([|M (t)| ≥ λ ]) ≤∫[|M(t)|≥λ ]

|M (t)|dP≤∫

Ω

|E (M (∞) |Ft)|dP

≤∫

Ω

E (|M (∞)| |Ft)dP =∫

Ω

|M (∞)|dP

and soP([|M (t)| ≥ λ ])≤ C

λ

From 62.10.37, this shows {M (t)} is uniformly integrable because this is true of the singlefunction |M (∞)|. By the submartingale convergence theorem, the convergence to M (∞)also takes place pointwise.

62.11 Hitting This Before ThatLet {M (t)} be a real valued martingale for t ∈ [0,T ] where T ≤ ∞ and M (0) = 0. In caseT = ∞, assume the conditions of Theorem 62.10.4 are satisfied. Thus there exists M (∞)and the M (t) are equiintegrable. With the Doob optional sampling theorem it is possible toestimate the probability that M (t) hits a before it hits b where a < 0 < b. There is no loss