808 CHAPTER 30. CONTINUOUS STOCHASTIC PROCESSES

Proof: If this is not so, there exists ε,δ > 0 and points of I,sn, tn such that even though|tn− sn|< 1

n ,P([∥X (sn)−X (tn)∥ ≥ ε])> δ . (30.1)

Taking a subsequence, still denoted by sn and tn there exists t ∈ I such that the above holdand limn→∞ sn = limn→∞ tn = t. Then

P([∥X (sn)−X (tn)∥ ≥ ε])

≤ P([∥X (sn)−X (t)∥ ≥ ε/2])+P([∥X (t)−X (tn)∥ ≥ ε/2]) .

But the sum of the last two terms converges to 0 as n→ ∞ by stochastic continuity of X att, violating 30.1 for all n large enough. ■

For a stochastically continuous process defined on a closed and bounded interval, therealways exists a measurable version. This is significant because then you can do things withproduct measure and iterated integrals.

Proposition 30.1.2 Let X be a stochastically continuous process defined on a closedinterval, I ≡ [a,b]. Then there exists a measurable version of X.

Proof: By Lemma 30.1.1 X is uniformly stochastically continuous and so there existsa sequence of positive numbers, {ρn} such that if |s− t|< ρn, then

P([∥X (t)−X (s)∥ ≥ 1

2n

])≤ 1

2n . (30.2)

Then let{

tn0 , t

n1 , · · · , tn

mn

}be a partition of [a,b] in which

∣∣tni − tn

i−1

∣∣< ρn. Now define Xn asfollows:

Xn (t)≡mn

∑i=1

X(tni−1)X[tn

i−1,tni )(t) , Xn (b)≡ X (b) .

Then Xn is obviously B(I)×F measurable because it is the sum of functions which are.Consider the set A on which {Xn (t,ω)} is a Cauchy sequence. This set is of the form

A = ∩∞n=1∪∞

m=1∩p,q≥m

[∥∥Xp−Xq∥∥< 1

n

]and so it is a B(I)×F measurable set. Now define

Y (t,ω)≡{

limn→∞ Xn (t,ω) if (t,ω) ∈ A0 if (t,ω) /∈ A

I claim Y (t,ω) = X (t,ω) for a.e. ω. To see this, consider 30.2. From the construction ofXn, it follows that for each t,

P([∥Xn (t)−X (t)∥ ≥ 1

2n

])≤ 1

2n (30.3)

Also, for a fixed t, if Xn (t,ω) fails to converge to X (t,ω) , then ω must be in infinitelymany of the sets,

Bn ≡[∥Xn (t)−X (t)∥ ≥ 1

2n

]