be a sequence of randomvariables such that Xnis ℱnmeasurable. Then XT
(ω)
≡ XT
(ω)
(ω)
is also arandom variable and it is measurable with respect to ℱT.
Proof: I assume the Xn have values in some topological space and each is
measurable because the inverse image of an open set is in ℱn. I need to show
XT−1
and so XT is ℱT is measurable as claimed. This proves the lemma.
Lemma 59.3.4Let S ≤ T be two stopping times such that T is bounded above and let
{Xn}
be a submartingale (martingale) adapted to the increasing sequence of σ algebras,
{ℱn}
. Then
E (XT|ℱS) ≥ XS
in the case where
{Xn }
is a submartingale and
E (X |ℱ ) = X
T S S
in the case where
{Xn }
is a martingale.
Proof: I will prove the case where
{Xn }
is a submartingale and note the
other case will only involve replacing ≥ with =. First recall that from Lemma
59.3.2ℱS⊆ℱT. Also let m be an upper bound for T. Then it follows from this
that
m ∫
E (|X |) = ∑ |X |dP < ∞
T i=1 [T=i] i
with a similar formula holding for E
(|XS |)
. Thus it makes sense to speak of
E
(XT |ℱS )
.
I need to show that if B ∈ℱS, so that B ∩
[S ≤ n]
∈ℱn for all n, then
∫ ∫
XTdP ≥ XSdP. (59.3.8)
B B
(59.3.8)
It suffices to do this for B of the special form
B = A∩ [S = i]
because if this is done, then the result follows from summing over all possible
values of S. Note that if B = A ∩
[S = m]
, then XT = XS = Xm and there is
nothing to prove in 59.3.8 so it can be assumed i ≤ m − 1. Then let B be of this
form.
is a martingale, you replace every occurance of ≥ in the above
argument with =. This proves the lemma.
This lemma is called the optional sampling theorem. Another version of this theorem
is the case where you have an increasing sequence of stopping times,
{Tn}
n=1∞. Thus if
{Xn}
is a sequence of random variables each ℱn measurable, the sequence
{XTn }
is also
a sequence of random variables such that XTn is measurable with respect to
ℱTn where ℱTn is an increasing sequence of σ fields. In the case where Xn is a
submartingale (martingale) it is reasonable to ask whether
{XTn }
is also a
submartingale (martingale). The optional sampling theorem says this is often the
case.
Theorem 59.3.5Let
{Tn}
be an increasing bounded sequence of stopping timesand let
{Xn }
be a submartingale (martingale) adapted to the increasing sequenceof σ algebras,
{ℱn}
. Then
{XTn}
is a submartingale (martingale) adapted to theincreasing sequence of σ algebras