62.7. DOOB OPTIONAL SAMPLING CONTINUOUS CASE 2083

∪t∈[a,a+ 1n ][X (t)> α]⊇ ∪t∈[a,a+ 1

n ],t∈Q[X (t)> α]

If ω is in the left side, then for some t in the given interval, X (t) > α. If for all s ∈[a,a+ 1

n

]∩Q you have X (s)≤ α, then you could take sn→ t where X (sn)≤ α and con-

clude from lower semicontinuity that X (t) ≤ α also. Thus there is some rational s whereX (s)> α and so the two sides are equal. Hence,

[τ = a] =(∩∞

n=m∪t∈[a,a+ 1n ],t∈Q

[X (t)> α])∩(∩t∈[0,a),t∈Q [X (t)≤ α]

)The first set on the right is in Fa+(1/m) and so is the next set on the right. Hence [τ = a] ∈∩mFa+(1/m) = Fa. To be a stopping time, one needs [τ ≤ a] ∈Fa. What of [τ < a]? Thisequals ∪t∈[0,a) [X (t)> α] = ∪t∈[0,a)∩Q [X (t)> α] ∈Fa, the equality following from lowersemicontinuity. Thus [τ ≤ a] = [τ = a]∪ [τ < a] ∈Fa.

Thus there do exist stopping times, the first hitting time above being an example. Whendealing with continuous stopping times on a normal filtration, one uses the following dis-crete stopping times

τn ≡∞

∑k=1

X[τ∈(tnk ,t

nk+1]]

tnk+1

where here∣∣tn

k − tnk+1

∣∣= rn for all k where rn→ 0. Then here is an important lemma.

Lemma 62.7.7 τn is a stopping time ([τn ≤ t] ∈Ft .) Also the inclusion Fτ ⊆Fτn holdsand for each ω,τn (ω) ↓ τ (ω).

Proof: Say t ∈ (tnk−1, t

nk ]. Then [τn ≤ t] =

[τ ≤ tn

k−1

]if t < tn

k and it equals[τ ≤ tn

k

]if

t = tnk . Either way [τn ≤ t] ∈Ft so it is a stopping time. Also from the definition, it follows

that τn ≥ τ and |τn (ω)− τ (ω)| ≤ rn which is given to converge to 0. Now suppose A∈Fτ

and say t ∈ (tnk−1, t

nk ] as above. Then

A∩ [τn ≤ t] = A∩[τ ≤ tn

k−1]∈Ftn

k−1⊆Ft if t < tn

k

andA∩ [τn ≤ t] = A∩ [τ ≤ tn

k ] ∈Ftnk= Ft if t = tn

k

Thus Fτ ⊆Fτn as claimed.Next is the claim that if X (t) is adapted to Ft , then X (τ) is adapted to Fτ just like the

discrete case.

Proposition 62.7.8 Let (Ω,F ,P) be a probability space and let σ ≤ τ be two stoppingtimes with respect to a filtration, Ft . Then Fσ ⊆Fτ . If X (t) is a right continuous stochas-tic process adapted to a normal filtration Ft and τ is a stopping time, ω → X (τ (ω)) isFτ measurable.

Proof: Let A ∈Fσ . Then A∩ [σ ≤ t] ∈Ft for all t ≥ 0. Since σ ≤ τ,

A∩ [τ ≤ t] =

∈Ft︷ ︸︸ ︷A∩ [σ ≤ t]∩ [τ ≤ t] ∈Ft