2076 CHAPTER 62. STOCHASTIC PROCESSES

with similar reasoning holding if A∈Fτ . In other words, if g is Fτ or Fk measurable, thenthe restriction of g to [τ = k] is measurable with respect to Fτ ∩ [τ = k] and Fk ∩ [τ = k] .Let Y be an arbitrary random variable in L1 (Ω,F ) . It follows, since A∩ [τ = k] is in bothFτ and Fk, ∫

A∩[τ=k]E (Y |Fτ)dP ≡

∫A∩[τ=k]

Y dP

≡∫

A∩[τ=k]E (Y |Fk)dP

Since this holds for an arbitrary set in Fτ ∩ [τ = k] = Fk ∩ [τ = k] , it follows

E (Y |Fτ) = E (Y |Fk) a.e. on [τ = k]

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.

62.6.2 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.

Theorem 62.6.5 Let {M (k)} be a martingale having values in E a separable real Banachspace with respect to the increasing sequence of σ algebras, {Fk} and let σ ,τ be twostopping times such that τ is bounded. Then M (τ) defined as

ω →M (τ (ω))

is integrable andM (σ ∧ τ) = E (M (τ) |Fσ ) .

Proof: By Proposition 62.6.3 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.