31.3. DOOB OPTIONAL SAMPLING CONTINUOUS CASE 835

Definition 31.3.1 A normal filtration is one which satisfies the following :

1. F0 contains all A ∈F such that P(A) = 0. Here F is the σ algebra which containsall Ft .

2. Ft = Ft+ for all t ∈ I where Ft+ ≡ ∩s>tFs.

For an F measurable [0,∞) valued function τ to be a stopping time, we want to havethe stopped process Xτ defined by Xτ (t)(ω)≡ X (t ∧ τ (ω))(ω) to be adapted whenever Xis right continuous and adapted. Thus a stopping time is a measurable function which canbe used to stop the process while retaining the property of being adapted. The definition ofsuch a condition which will make τ a stopping time is the same as in the case of a discreetprocess.

Definition 31.3.2 τ an F measurable function is a stopping time if [τ ≤ t] ∈Ft .

Then this definition does what is desired. This is in the following proposition. Forconvenience, here is a definition.

Definition 31.3.3 Let {t}k ≡ 2−kn where n is as large as possible and have 2−kn≤t.

It seems like the theory is based on reducing to discrete stopping times defined asfollows.

Definition 31.3.4

τk (ω)≡∞

∑n=0

Xτ−1((n2−k,(n+1)2−k]) (ω)(n+1)2−k.

Thus τk has values in the set{

n2−k}∞

n=0 ,τk ≥ τ and τk is within 2−k of τ .

Then τk is a discrete stopping time with respect to the increasing σ algebras F{t}k .This is in the following lemma.

Lemma 31.3.5 Let τ be a stopping time and let τk be defined above in Definition 31.3.4.Then [τk ≤ {t}k] ∈F{t}k and for all t, [τk ≤ t] ∈Ft . If you have finitely many stoppingtimes

{σ k}n

k=1 for n < ∞ then σ ≡min{

σ k}n

k=1 and σ ≡max{

σ k}n

k=1 are also stoppingtimes.

Proof: If t = (n+1)2−k then [τk ≤ {t}k] is the same as[τ ≤ (n+1)2−k

]= [τ ≤ t] ∈

Ft =F{t}k . The other case is where for some n, t ∈(n2−k,(n+1)2−k

). Then {t}k = n2−k

and so [τk ≤ {t}k] is[τ ≤ n2−k

]∈Fn2−k = F{t}k . Thus, in particular, [τk ≤ t] ∈Ft since

{t}k ≤ t.As to the second claim,[

min{

σk}n

k=1≤ t]= ∪n

k=1

[σ

k ≤ t]∈Ft

and[max

{σ k}n

k=1 ≤ t]= ∩n

k=1

[σ k ≤ t

]∈Ft . ■