796 CHAPTER 29. MARTINGALES

Note this is equivalent to saying [τ = n] ∈Fn because [τ = n] = [τ ≤ n]\ [τ ≤ n−1] . Forτ a stopping time define Fτ as follows.

Fτ ≡ {A ∈F : A∩ [τ ≤ n] ∈Fn for all n ∈ N} (29.10)

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

Lemma 29.4.2 The requirement 29.10 is equivalent to saying that A∩ [τ = n] ∈Fn forall n ∈ N.

Proof: [τ = n]∩A = [τ ≤ n]∩A \ [τ ≤ n−1]∩A ∈ Fn so if 29.10 holds, then A∩[τ = n] ∈Fn for all n. Conversely, if A∩ [τ = n] ∈Fn, then A∩ [τ ≤ n] = ∪k≤nA∩ [τ = k]which is the union of sets in Fn since the Fk are increasing in k. ■

Another good thing to observe is the following lemma.

Lemma 29.4.3 If σ ≤ τ and σ ,τ are two stopping times, then Fσ ⊆Fτ .

Proof: Say A ∈Fσ which means that A∩ [σ ≤ n] ∈Fn for all n. Now consider A∩[τ = n] . Is this in Fn? Since σ ≤ τ,

A∩ [τ = n] =∈Fn

[τ = n]∩∈Fn

∪nj=1A∩ [σ = j] ∈Fn

since each [σ = j]∩A ∈Fn. ■Next is a significant observation that a stochastic process A(n) where A(n) is Fn mea-

surable satisfies A(τ) ∈Fτ .

Proposition 29.4.4 Let A(k) be Fk measurable for each k where Fk is an increasingsequence of σ algebras. Then A(τ) is Fτ measurable if τ is a stopping time correspondingto the Fk. If σ ,τ are two stopping times, then so are τ ∧σ and τ ∨σ , the minimum andmaximum of the two stopping times.

Proof: I need to show that for O an open set [A(τ) ∈ O] is Fτ measurable. I need toshow that [A(τ) ∈ O]∩ [τ ≤ k]∈Fk for each k. It suffices to show that for each j, [τ = j]∩[A( j) ∈ O]∩ [τ ≤ k]∈Fk for each k. If j≤ k, then left side reduces to [τ = j]∩ [A( j) ∈ O] .However, A( j) is F j measurable and so [A( j) ∈ O] ∈F j ⊆Fk while [τ = j] ∈F j ⊆Fkso all is well if j≤ k. However, if j > k, then the expression [τ = j]∩ [A( j)≤ r]∩ [τ ≤ k] =/0 ∈Fk and so it works in this case also. Thus A(τ) is indeed Fτ measurable.

For the last claim, [τ ∧σ ≤ j] = [τ ≤ j]∪ [σ ≤ j] ∈ F j and [τ ∨σ ≤ j] = [τ ≤ j]∩[σ ≤ j] ∈F j. ■

Example 29.4.5 As an example of a stopping time, let Xn be Fn measurable where theFn are increasing σ algebras. Let O be a Borel set, and let τ (ω) be the first n such thatXn (ω) ∈O. If X−1

n (O) = /0 for all n, then τ (ω)≡ ∞ and we consider F∞ to be F . This isan example of a stopping time called the first hitting time. With these discreet processes, itis enough to let O be Borel.

Lemma 29.4.6 The first hitting time of a Borel set O is a stopping time. Also Fτ is a σ

algebra.