61.6.2 Doob Optional Sampling Theorem
With this lemma, here is a major theorem, the optional sampling theorem of Doob. This
one is for martingales having values in a Banach space. To begin with, consider the case
of a martingale defined on a countable set.
Theorem 61.6.5 Let
be a martingale having values in E a separable
real Banach space with respect to the increasing sequence of σ algebras,
and let σ,τ be two stopping times such that τ is bounded. Then M
is integrable and
Proof: By Proposition 61.6.3 M
Next note that since τ is bounded by some l,
This proves the first assertion and makes possible the consideration of conditional
Let l ≥ τ as described above. Then for k ≤ l, by Lemma 61.6.4,
implying that if g is either ℱk measurable or ℱτ measurable, then its restriction to
measurable and so if A ∈ℱk ∩
Therefore, since A
k ≤ l.
since it is true on each
k ≤ l.
Now consider E
on the set
. By Lemma
, on this
If j ≤ i, this reduces to
If j > i, this reduces to
and since this exhausts all possibilities for values of σ and τ, it follows
You can also give a version of the above to submartingales. This requires the following
very interesting decomposition of a submartingale into the sum of an increasing
stochastic process and a martingale.
Theorem 61.6.6 Let
be a submartingale. Then there exists a unique
,A1 = 0
- An is ℱn−1 adapted for all n ≥ 1 where ℱ0 ≡ℱ1.
and also Xn = Mn + An.
Proof: Let A1 ≡ 0 and define
It follows An is ℱn−1 measurable. Since
is a submartingale,
It is a submartingale because
Now let Mn
be defined by
Then from 61.6.25,
This proves the existence part.
It remains to verify uniqueness. Suppose then that
both satisfy the conditions of the theorem and
are both martingales. Then
and so, since An′− An is ℱn−1 measurable and
is a martingale,
Continuing this way shows Mn−Mn′
is a constant. However, since A1′−A1
= 0 = M1 −M1′,
it follows Mn
and this proves uniqueness. ■
Now here is a version of the optional sampling theorem for submartingales.
Theorem 61.6.7 Let
be a real valued submartingale with respect to the
increasing sequence of σ algebras,
and let σ ≤ τ be two stopping times such that τ
is bounded. Then M
is integrable and
Without assuming σ ≤ τ, one can write
Proof: That ω → X
is integrable follows right away as in the optional
sampling theorem for martingales. You just consider the finitely many values of
Use Theorem 61.6.6 above to write
where M is a martingale and A is increasing with A
= 0 as discussed in Theorem
Now since A is increasing, you can use the optional sampling theorem for martingales,
Theorem 61.6.5 to conclude that, since ℱσ∧τ ⊆ℱσ and A
In summary, the main results for stopping times for discrete filtrations are the
following definitions and theorems.
This last theorem implies the following amazing result. From these fundamental
properties, we obtain the optional sampling theorem for martingales and submartingales.