832 CHAPTER 31. OPTIONAL SAMPLING THEOREMS

To see this,

(Y ◦ τ)−1 (G)∩ [τ ≤ n] = ∪k

∈Fk︷ ︸︸ ︷

Y (k)−1 (G)

∩ [τ = k]∩ [τ ≤ n]

The term in the union is /0 if k > n and so the whole thing reduces to

∪nk=1

∈Fk︷ ︸︸ ︷

Y (k)−1 (G)

∩ [τ = k] ∈Fn

showing that (Y ◦ τ)−1 (G) ∈Fτ .The following lemma contains the fundamental properties of stopping times for discrete

filtrations. It was Lemma 29.4.7.

Lemma 31.1.4 In the situation of Definition 31.1.1,

1. Fτ ∩ [τ = i] = Fi∩ [τ = i] and E (X |Fτ) = E (X |Fi) a.e. on the set [τ = i] . Also ifA ∈Fτ or Fi, then A∩ [τ = i] ∈Fi∩Fτ .

2. E (X |Fτ) = E (X |Fi) a.e. on the set [τ ≤ i] .

3. Also, if σ ≤ τ, then Fσ ⊆Fτ

Proof: The first two are in the above mentioned lemma. The first part of 1. comesfairly quickly from the definition. The next part of 1. about the conditional expectationsis essentially because one can regard Fτ ∩ [τ = i] and Fi∩ [τ = i] as two equal σ algebrascontained in [τ = i] and so the two conditional expectations are the same on [τ = i]. Thethird part of 1. also follows from the definition. Then 2. is clearly true from 1. applied to[τ = j] for j ≤ i.

Say A ∈Fτ . Then for j ≤ i, [τ = j] ∈Fτ because [τ = j]∩ [τ ≤ k] ∈Fk for each k.Thus ∫

A∩[τ= j]XdP =

∫A∩[τ= j]

E (X |Fτ)dP =∫

A∩[τ= j]∈F j

E (X |F j)dP

=∫

A∩[τ= j]E (X |Fi)dP

Since A is arbitrary, E (X |Fτ) = E (X |Fi).Now consider 3. If A ∈Fσ , this means A∩ [σ ≤ i] ∈Fi or equivalently, A∩ [σ = i] ∈

Fi for all i. Take such an A. Then A∩ [τ = n] = ∪ni=1A∩ [σ = i] ∈Fn and so Fσ ⊆Fτ . ■

The assertion thatE (Y |Fτ) = E (Y |Fk) a.e.

on [τ = k] and that a function g which is Fτ or Fk measurable when restricted to [τ = k]is G measurable for

G =[τ = k]∩Fτ = [τ = k]∩Fk

is the main result in the above lemma and this fact leads to the amazing Doob optionalsampling theorem below. Also note that if Y (k) is any process defined on the positiveintegers k, then by definition, Y (k)(ω) = Y (τ (ω))(ω) on the set [τ = k] because τ isconstant on this set.