62.7. DOOB OPTIONAL SAMPLING CONTINUOUS CASE 2081

Definition 62.7.2 τ an F measurable function is a stopping time if [τ ≤ t] ∈Ft .

What follows will be more discussion of this simple idea of preserving the process ofbeing adapted when the process is stopped.

Then the above discussion shows the following proposition.

Proposition 62.7.3 Let {Ft} be a normal filtration and let X (t) be a right continuousprocess adapted to {Ft} . Then if τ is a stopping time, it follows that the stopped processXτ defined by Xτ (t)≡ X (τ ∧ t) is also adapted.

Definition 62.7.4 Let (Ω,F ,P) be a probability space and let Ft be a filtration. A mea-surable function, τ : Ω→ [0,∞] is called a stopping time if

[τ ≤ t] ∈Ft

for all t ≥ 0. Associated with a stopping time is the σ algebra, Fτ defined by

Fτ ≡ {A ∈F : A∩ [τ ≤ t] ∈Ft for all t} .

These sets are also called those “prior” to τ.

Note that Fτ is obviously closed with respect to countable unions. If A ∈Fτ , then

AC ∩ [τ ≤ t] = [τ ≤ t]\ (A∩ [τ ≤ t]) ∈Ft

Thus Fτ is a σ algebra.

Proposition 62.7.5 Let B be an open subset of topological space E and let X (t) be a rightcontinuous Ft adapted stochastic process such that Ft is normal. Then define

τ (ω)≡ inf{t > 0 : X (t)(ω) ∈ B} .

This is called the first hitting time. Then τ is a stopping time. If X (t) is continuous andadapted to Ft , a normal filtration, then if H is a nonempty closed set such that H =∩∞

n=1Bnfor Bn open, Bn ⊇ Bn+1,

τ (ω)≡ inf{t > 0 : X (t)(ω) ∈ H}

is also a stopping time.

Proof: Consider the first claim. ω ∈ [τ = a] implies that for each n ∈ N, there existst ∈[a,a+ 1

n

]such that X (t) ∈ B. Also for t < a, you would need X (t) /∈ B. By right

continuity, this is the same as saying that X (d) /∈ B for all rational d < a. (If t < a, youcould let dn ↓ t where X (dn) ∈ BC, a closed set. Then it follows that X (t) is also in theclosed set BC.) Thus, aside from a set of measure zero, for each m ∈ N,

[τ = a] =(∩∞

n=m∪t∈[a,a+ 1n ][X (t) ∈ B]

)∩(∩t∈[0,a)

[X (t) ∈ BC])