62.3. FILTRATIONS 2067

Proposition 62.3.13 Let P denote the predictable σ algebra and let R denote the pro-gressively measurable σ algebra. Then P ⊆R.

Proof: Let G denote those sets of P such that they are also in R. Then G clearlycontains the π system of sets {0}×A,A ∈F0, and (s, t]×A,A ∈Fs. Furthermore, G isclosed with respect to countable disjoint unions and complements. It follows G containsthe σ algebra generated by this π systems which is P . This proves the proposition.

Proposition 62.3.14 Let X (t) be a stochastic process having values in E a complete metricspace and let it be Ft adapted and left continuous. Then it is predictable. Also, if X (t) isstochastically continuous and adapted on [0,T ] , then it has a predictable version.

Proof:Define Im,k ≡ ((k−1)2−mT,k2−mT ] if k≥ 1 and Im,0 = {0} if k = 1. Then define

Xm (t) ≡2m

∑k=1

X(T (k−1)2−m)X((k−1)2−mT,k2−mT ] (t)

+X (0)X[0,0] (t)

Here the sum means that Xm (t) has value X (T (k−1)2−m) on the interval

((k−1)2−mT,k2−mT ].

Thus Xm is predictable because each term in the sum is. Thus

X−1m (U) = ∪2m

k=1(X(T (k−1)2−m)X((k−1)2−mT,k2−mT ]

)−1(U)

= ∪2m

k=1((k−1)2−mT,k2−mT ]×(X(T (k−1)2−m))−1

(U) ,

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

X (t,ω) = limm→∞

Xm (t,ω)

and so X is predictable.Next consider the other claim. Since X is stochastically continuous on [0,T ] , it is

uniformly stochastically continuous on this interval by Lemma 62.1.1. Therefore, thereexists a 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 (62.3.19)

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