1960 CHAPTER 60. CONDITIONAL, MARTINGALES

Proof: This follows from Lemma 60.4.3

Example 60.4.5 Let {Xn} be a sequence of real random variables such that Xn is Fnmeasurable and let A be a Borel subset of R. Let T (ω) denote the first time Xn (ω) is in A.Then T is a stopping time. It is called the first hitting time.

To see this is a stopping time,

[T ≤ l] = ∪li=1X−1

i (A) ∈Fl .

60.5 Optional Stopping Times And Martingales60.5.1 Stopping Times And Their PropertiesThe purpose of this section is to consider a special optional sampling theorem for martin-gales which is superior to the one presented earlier. I have presented a different treatmentof the fundamental properties of stopping times also. See Kallenberg [77] for more.

Definition 60.5.1 Let (Ω,F ,P) be a probability space and let {Fn}∞

n=1 be an increasingsequence of σ algebras each contained in F . A stopping time is a measurable function, τ

which maps Ω to N,τ−1 (A) ∈F for all A ∈P (N) ,

such that for all n ∈ N,[τ ≤ n] ∈Fn.

Note this is equivalent to saying[τ = n] ∈Fn

because[τ = n] = [τ ≤ n]\ [τ ≤ n−1] .

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 .

First note that for τ a stopping time, Fτ is a σ algebra. This is in the next proposition.

Proposition 60.5.2 For τ a stopping time, Fτ is a σ algebra and if Y (k) is Fk measurablefor all k, 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