62.6. OPTIONAL SAMPLING THEOREMS 2079

Theorem 62.6.7 Let {X (k)} be a real valued submartingale with respect to the increasingsequence of σ algebras, {Fk} and let σ ≤ τ be two stopping times such that τ is bounded.Then M (τ) defined as

ω → X (τ (ω))

is integrable andX (σ)≤ E (X (τ) |Fσ ) .

Without assuming σ ≤ τ, one can write

X (σ ∧ τ)≤ E (X (τ) |Fσ )

Proof: That ω→ X (τ (ω)) is integrable follows right away as in the optional samplingtheorem for martingales. You just consider the finitely many values of τ .

Use Theorem 62.6.6 above to write

X (n) = M (n)+A(n)

where M is a martingale and A is increasing with A(n) being Fn−1 measurable and A(0) =0 as discussed in Theorem 62.6.6. Then

E (X (τ) |Fτ) = E (M (τ)+A(τ) |Fσ )

Now since A is increasing, you can use the optional sampling theorem for martingales,Theorem 62.6.5 to conclude that, since Fσ∧τ ⊆Fσ and A(σ ∧ τ) is Fσ∧τ measurable,

≥ E (M (τ)+A(σ ∧ τ) |Fσ ) = E (M (τ) |Fσ )+A(σ ∧ τ)

= M (σ ∧ τ)+A(σ ∧ τ) = X (σ ∧ τ) .

In summary, the main results for stopping times for discrete filtrations are the followingdefinitions and theorems.

[τ ≤ m] ∈Fm

A ∈Fτ means A∩ [τ ≤ m] ∈Fm for any m

X adapted implies X (τ) is Fτ measurable

Fσ∧τ = Fσ ∩Fτ

[τ = k]∩Fk = [τ = k]∩Fτ

This last theorem implies the following amazing result. From these fundamental properties,we obtain the optional sampling theorem for martingales and submartingales.

E (Y |Fτ) = E (Y |Fk) a.e. on [τ = k]