31.2. REVIEW OF DOOB OPTIONAL SAMPLING THEOREM 833

31.2 Review of Doob Optional Sampling TheoremWith this lemma, here is a major theorem, the optional sampling theorem of Doob. Thisone is for martingales having values in a Banach space. To begin with, consider the case ofa martingale defined on a countable set. This was discussed earlier but it is the sort of thingthat seems to me should be repeated because it is so amazing.

Theorem 31.2.1 Let {M (k)} be a martingale having values in E a separable realBanach space with respect to the increasing sequence of σ algebras, {Fk} and let σ ,τbe two stopping times such that τ is bounded. Then M (τ) defined as ω → M (τ (ω)) isintegrable and

M (σ ∧ τ) = E (M (τ) |Fσ ) .

Proof: By Proposition 31.1.3 M (τ) is Fτ measurable.Next note that since τ is bounded by some l,∫

Ω

∥M (τ (ω))∥dP≤l

∑i=1

∫[τ=i]∥M (i)∥dP < ∞.

This proves the first assertion and makes possible the consideration of conditional expecta-tion.

(E (M (l) |Fτ) = M (τ)) Let l≥ τ as described above. Then for k≤ l, by Lemma 31.1.4,

Fk ∩ [τ = k] = Fτ ∩ [τ = k]≡ G

implying that if g is either Fk measurable or Fτ measurable, then its restriction to [τ = k]is G measurable and so if A ∈Fτ ∩ [τ = k] then∫

AE (M (l) |Fτ)dP ≡

∫A

M (l)dP =∫

AE (M (l) |Fk)dP

=∫

AM (k)dP =

∫A

M (τ)dP (on A,τ = k)

Therefore, since A was arbitrary, E (M (l) |Fτ) = M (τ) a.e. on [τ = k] for every k ≤ l. Itfollows E (M (l) |Fτ) = M (τ) a.e. since it is true on each [τ = k] for all k ≤ l.

(M (σ ∧ τ) = E (M (τ) |Fσ )) Now consider E (M (τ) |Fσ ) on the set [σ = i]∩ [τ = j].By Lemma 31.1.4, on this set,

E (M (τ) |Fσ ) = E (M (τ) |Fi) = E (E (M (l) |Fτ) |Fi) = E (E (M (l) |F j) |Fi)

If j ≤ i, this reduces to

E (M (l) |F j) = M ( j) = M (σ ∧ τ) .

If j > i, this reduces toE (M (l) |Fi) = M (i) = M (σ ∧ τ)

and since this exhausts all possibilities for values of σ and τ, it follows

E (M (τ) |Fσ ) = M (σ ∧ τ) a.e. ■

You can also give a version of the above to sub-martingales. This requires the followingvery interesting decomposition of a sub-martingale into the sum of an increasing stochasticprocess and a martingale. This was presented earlier as Lemma 29.4.9.