2086 CHAPTER 62. STOCHASTIC PROCESSES

Lemma 62.7.12 Let X (t) be a right continuous nonnegative submartingale such that thefiltration {Ft} is normal. Recall this includes

Ft = ∩s>tFs.

Also let τ be a stopping time with values in [0,T ] . Let Pn ={

tnk

}mn+1k=1 be a sequence of

partitions of [0,T ] which have the property that

Pn ⊆Pn+1, limn→∞||Pn||= 0,

where||Pn|| ≡ sup

{∣∣tnk − tn

k+1∣∣ : k = 1,2, · · · ,mn

}Then let

τn (ω)≡mn

∑k=0

tnk+1Xτ−1((tn

k ,tnk+1])

(ω)

It follows that τn is a stopping time and also the functions |X (τn)| are uniformly integrable.Furthermore, |X (τ)| is integrable.

Proof: First of all, say t ∈ (tnk , t

nk+1]. If t < tn

k+1, then

[τn ≤ t] = [τ ≤ tnk ] ∈Ftn

k⊆Ft

and if t = tnk+1, then [

τn ≤ tnk+1]=[τ ≤ tn

k+1]∈Ftn

k+1= Ft

and so τn is a stopping time. It follows from Proposition 62.7.8 that X (τn) is Fτn measur-able.

Now from Lemma 60.4.3 or Theorem 62.6.7, X (0) ,X (τn) ,X (T ) is a submartingale.Then ∫

[X(τn)≥λ ]X (τn)dP ≤

∫[X(τn)≥λ ]

E (X (T ) |Fτn)dP

=∫

Ω

E(X[X(τn)≥λ ]X (T ) |Fτn

)dP

=∫[X(τn)≥λ ]

X (T )dP

From maximal estimates, for example Theorem 60.2.8,

P([X (τn)≥ λ ])≤ 1λ

∫Ω

X (T )+ dP =1λ

∫Ω

X (T )dP

and now it follows from the above that the random variables X (τn) are equiintegrable.Recall this means that

limλ→∞

supn

∫[X(τn)≥λ ]

X (τn)dP = 0

Hence they are uniformly integrable.