2440 CHAPTER 72. THE HARD ITO FORMULA

a finite union of predictable sets. Since X is left continuous,

X (t,ω) = limm→∞

Xm (t,ω)

and so X is predictable.Now suppose that for ω /∈ N, P(N) = 0, t→ X (t,ω) is continuous. Then applying the

above argument to X (t)XNC it follows X (t)XNC is predictable by completeness of Ft ,X (t)XNC is Ft measurable.

Next consider the other claim. Since X is stochastically continuous on [0,T ] it is uni-formly stochastically continuous on this interval by Lemma 62.1.1. Therefore, there existsa sequence of partitions of [0,T ] , the mth being

0 = tm,0 < tm,1 < · · ·< tm,n(m) = T

such that for Xm defined as above, then for each t

P([

d (Xm (t) ,X (t))≥ 2−m])≤ 2−m (72.1.1)

Then as above, Xm is predictable. Let A denote those points of PT at which Xm (t,ω)converges. Thus A is a predictable set because it is just the set where Xm (t,ω) is a Cauchysequence. Now define the predictable function Y

Y (t,ω)≡{

limm→∞ Xm (t,ω) if (t,ω) ∈ A0 if (t,ω) /∈ A

From 72.1.1 it follows from the Borel Cantelli lemma that for fixed t, the set of ω whichare in infinitely many of the sets,[

d (Xm (t) ,X (t))≥ 2−m]has measure zero. Therefore, for each t, there exists a set of measure zero, N (t) such thatfor ω /∈ N (t) and all m large enough[

d (Xm (t,ω) ,X (t,ω))< 2−m]Hence for ω /∈ N (t) , (t,ω) ∈ A and so Xm (t,ω)→ Y (t,ω) which shows

d (Y (t,ω) ,X (t,ω)) = 0 if ω /∈ N (t) .

The predictable version of X (t) is Y (t).Finally consider the claim about the specific example where

X ∈C ([0,T ] ;Lp (Ω;F)) .

P([||X (t)−X (s)||F ≥ ε])εp ≤

∫Ω

||X (t)−X (s)||pF dP≤ εpδ

provided |s− t| sufficiently small. Thus

P([||X (t)−X (s)||F ≥ ε])< δ

when |s− t| is small enough.