Chapter 31

Optional Sampling TheoremsAs with discreet martingales, there is a notion of stopping time and optional samplingtheorems. These are considered by approximating with discreet stopping times. It is likethe case of the integral where one uses step functions or simple functions to approximate agiven function.

31.1 Review of Discreet Stopping TimesFirst it is necessary to define the notion of a stopping time. The following definition wasdiscussed earlier in the context of discreet processes.

Definition 31.1.1 Let (Ω,F ,P) be a probability space and let {Fn}∞

n=1 be an in-creasing sequence of σ algebras each contained in F , called a discrete filtration. A stop-ping 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 .

It is clear that Fτ is a σ algebra.The most important example of a stopping time is the first hitting time.

Example 31.1.2 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.

This led to the following proposition. It was Proposition 29.4.4.

Proposition 31.1.3 For τ a stopping time, Fτ is a σ algebra and if Y (k) is Fk mea-surable for all k,Y (k) having values in a separable Banach space E, then

ω → Y (τ (ω))

is Fτ measurable.

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