62.6. OPTIONAL SAMPLING THEOREMS 2073

For τ a stopping time define Fτ as follows.

Fτ ≡ {A ∈F : A∩ [τ ≤ n] ∈Fn for all n ∈ N}

These sets in Fτ are referred to as “prior” to τ .

The most important example of a stopping time is the first hitting time.

Example 62.6.2 The first hitting time of an adapted process X (n) of a Borel set G is astopping time. This is defined as

τ ≡min{k : X (k) ∈ G}

To see this, note that

[τ = n] = ∩k<n[X (k) ∈ GC]∩ [X (n) ∈ G] ∈Fn.

Proposition 62.6.3 For τ a stopping time, Fτ is a σ algebra and if Y (k) is Fk measurablefor all k,Y (k) having values in a separable Banach space E, then

ω → Y (τ (ω))

is Fτ measurable.

Proof: Let An ∈Fτ . I need to show ∪nAn ∈Fτ . In other words, I need to show that

∪nAn∩ [τ ≤ k] ∈Fk

The left side equals∪n (An∩ [τ ≤ k])

which is a countable union of sets of Fk and so Fτ is closed with respect to countableunions. Next suppose A ∈Fτ .(

AC ∩ [τ ≤ k])∪ (A∩ [τ ≤ k]) = Ω∩ [τ ≤ k]

and Ω∩ [τ ≤ k] ∈Fk. Therefore, so is AC ∩ [τ ≤ k] .It remains to verify the last claim. Let B be an open set in E. Is

[Y (τ) ∈ B] ∈Fτ ?

Is[Y (τ) ∈ B]∩ [τ ≤ k] ∈Fk for all k?

This equals

∪ki=1 [Y (τ) ∈ B]∩ [τ = i] = ∪k

i=1 [Y (i) ∈ B]∩ [τ = i] ∈Fk

Therefore, Y (τ) must be Fτ measurable.The following lemma contains the fundamental properties of stopping times for discrete

filtrations.