Stochastic Analysis in Mathematical Finance
MA5248
Lecture Notes 2
Conditional Expectations
¸1. Two Examples.
Let (Ω, T, IP) be a probability space. We are going to introduce “conditional
expectations”. Let us begin with some special cases and then proceed to two
concrete examples (in items 3 and 6).
1. Let Λ ∈ T with IP(Λ) 0. Deﬁne IP
Λ
() on T as follows:
IP
Λ
(E) =
IP(Λ ∩ E)
IP(Λ)
. (1.1)
It is trivial to check that IP
Λ
() thus deﬁned in (1.
Stochastic Analysis in Mathematical FinanceMA5248Lecture Notes 2Conditional Expectations
§
1. Two Examples.
Let (Ω
,
F
,
IP) be a probability space. We are going to introduce “
conditionalexpectations
”. Let us begin with some special cases and then proceed to twoconcrete examples (in items 3 and 6).1. Let Λ
∈ F
with IP(Λ)
>
0. Deﬁne IP
Λ
(
·
) on
F
as follows:IP
Λ
(
E
) =IP(Λ
∩
E
)IP(Λ)
.
(1
.
1)It is trivial to check that IP
Λ
(
·
) thus deﬁned in (1.1) is a probability measure (p.m.)on
F
, which is known as the “conditional probability relative to Λ.” It is usuallywritten as IP(
·
Λ). Note also that (1.1) can also be written asIP(Λ
∩
E
) = IP(Λ)IP
Λ
(
E
)
.
The integral w.r.t. this p.m. is called the “conditional expectation relative to Λ”:
EE
Λ
[
X
] =
Ω
X
(
ω
)IP
Λ
(
dω
) =1IP(Λ)
Λ
X
(
ω
)IP(
dω
) =
EE
[
X
; Λ]IP(Λ)
,
(1
.
2)provided that
EE
[
X
] is deﬁned, (i.e.,
EE

X

<
∞
).(To establish (1.2), one only needs to make use of the “
standard machine
,” namely,to begin with
X
as indicator r.v.’s; then simple r.v.’s; nonnegative r.v.’s; etc. Fordetails, see
§
3.2 below.) Recall that
EE
[
X
; Λ] =
EE
[
XI
Λ
] =
Λ
X d
IP
.
2. However, if IP(Λ) = 0, we decree that IP
Λ
(
E
) = 0 for every
E
∈ F
.3. We now give our ﬁrst example.
Example 1.
Suppose there is a countable measurable partition
{
Λ
n
, n
≥
1
}
of Ω,namely:Ω =
∪
n
Λ
n
,
Λ
n
∈ F
,
Λ
m
∩
Λ
n
=
∅
,
if
m
=
n.
1
Then we have, when
EE
[
X
] is deﬁned, (i.e.,
EE

X

<
∞
),IP(
E
) =
n
IP(Λ
n
∩
E
) =
n
IP(Λ
n
)IP
Λ
n
(
E
)
EE
[
X
] =
n
Λ
n
X
(
ω
)IP(
dω
) =
n
IP(Λ
n
)
EE
Λ
n
[
X
]
,
(1
.
3)Let
G
be the
σ
algebra generated by this partition. Given an integrable r.v.
X
∈ F
,we deﬁne the
function
, denoted by
EE
[
X
 G
] (or
EE
G
[
X
]), on Ω by: for
ω
∈
Ω,
EE
[
X
 G
](
ω
) =
n
EE
Λ
n
[
X
]
I
Λ
n
(
ω
)
.
Observe that
EE
[
X
 G
] is a discrete r.v. that assumes the value
EE
Λ
n
[
X
] =
EE
[
X
; Λ
n
]IP(Λ
n
)on the event Λ
n
for each
n
. More precisely, the above function
EE
[
X
G
] can beexpressed as
EE
[
X
 G
](
·
) =
n
EE
Λ
n
[
X
]
I
Λ
n
(
·
) =
n
EE
[
X
; Λ
n
]IP(Λ
n
)
I
Λ
n
(
·
)
.
(1
.
4)Obviously, this function
EE
[
X
 G
] is measurable with respect to the sub
σ
algebra
G
, i.e.,
EE
[
X
 G
]
∈ G
.Now (1.3) can be rewritten as
EE
[
X
] =
Ω
X d
IP =
n
Λ
n
EE
[
X
G
]
d
IP =
Ω
EE
[
X
G
]
d
IP
.
(1
.
5)Furthermore, if Λ
∈ G
, then Λ is a union of a subcollection of the Λ
n
’s, and notethat the following also holds:
∀
Λ
∈ G
:
EE
[
X
; Λ] =
EE
[
XI
Λ
] =
Λ
X d
IP =
Λ
EE
[
X
 G
]
d
IP
.
(1
.
6)(Clearly, (1.5) is a special case of (1.6) with Λ being replaced by Ω.)4.
Remarks:
(a) Look at (1.6) and especially note that the last two terms are integrals over thesame event Λ
∈ G
; but the integrand
X
of the 3rd term is measurable w.r.t.
F
,2
while the integrand
EE
[
X
G
] on the far right is measurable w.r.t. the sub
σ
algebra
G
.(b) If Λ in (1.6) is replaced by another event in
F \G
, then (1.6) may not hold anymore. In other words, (1.6) holds in general
for
Λ
∈ G
only
.(c) We now claim that for any integrable
X
,
EE
[
X
 G
] is
unique
a.s. That is, if thereis another r.v.
W
∈ G
such that
∀
Λ
∈ G
:
Λ
X d
IP =
Λ
W d
IP
,
(1
.
7)then
W
=
EE
[
X
G
] a.s.
Proof of the Claim.
PutΛ =
{
ω
:
EE
[
X
 G
](
ω
)
> W
(
ω
)
}
.
Obviously, Λ
∈ G
, and Λ =
∪
n
Λ
n
, whereΛ
n
=
{
ω
:
EE
[
X
 G
](
ω
)
> W
(
ω
) +
n
−
1
}
.
If IP(Λ)
>
0, then there exists an
n
≥
1 such that IP
{
Λ
n
}
>
0. On this Λ
n
,
Λ
n
X d
IP =
Λ
n
W d
IP
<
Λ
n
EE
[
X
 G
]
d
IP =
Λ
n
X d
IP
,
a contradiction. So, IP(Λ) = 0 and
EE
[
X
 G
]
≤
W
a.s. In a similar fashion, byinterchanging
EE
[
X
 G
] and
W
above we arrive at
W
≤
EE
[
X
 G
] a.s., and henceconclude that
W
=
EE
[
X
G
] a.s., which is the claim.5. We have therefore proved that the
EE
[
X
 G
] in (1.6) is unique up to an equivalence.From now on we will also use
EE
[
X
 G
] to denote the corresponding equivalenceclass, and call any particular member of the class a “
version
” of the
conditionalexpectation relative to the
σ
algebra
G
.
Before proceeding, let us summarize what we have obtained. For a countable measurable partition
{
Λ
n
n
≥
1
}
, let
G
=
σ
{
Λ
n
n
≥
1
} ⊂ F
. For
X
∈ F
,
EE
[
X
 G
]
∈ G
and it is unique a.s.; and if
X
has ﬁnite mean, then
EE
[
X
 G
] has also ﬁnite mean(by(1.5)).6. Now the
second
example.
Example 2.
Let
Z
be a discrete r.v. taking distinct values
a
1
, a
2
, a
3
, ...
. PutΛ
n
=
{
Z
=
a
n
}
. Then we have a countable measurable partition
{
Λ
n
, n
≥
1
}
of 3
Ω. It is obvious to see that
σ
(
Z
) =
σ
{
Λ
n
, n
≥
1
}
, denoted by
G
. By what hasbeen mentioned above in the ﬁrst example,
EE
[
X
G
] is deﬁned for any
X
∈ F
with
EE

X

<
∞
, and then we can deﬁne the
conditional expectation relative to ther.v.
Z
to be
EE
[
X

Z
]
def.
=
EE
[
X
 G
] =
EE
[
X

σ
(
Z
)]
.
Obviously, the expectation of
X
may change given that the event Λ
n
has occurred.Indeed, if Q(Λ)
def.
= IP
{
Λ

Λ
n
}
,
it makes more sense to calculate
EE
Q[
X
] than it does to calculate
EE
IP
[
X
]. (
EE
IP
{·}
denotes expectation with respect to the probability measure IP.)An important observation that we would like to mention is that:
EE
[
X

Z
] is nowa
function
of
Z
, i.e., there is a Borel measurable function
ϕ
such that
EE
[
X

Z
] =
ϕ
(
Z
) a.s.More precisely, deﬁne
ϕ
(
n
) =
EE
[
X
; Λ
n
)IP(Λ
n
)=
EE
[
X
;
Z
=
n
]IP(
Z
=
n
)
.
(1
.
8)For
ω
∈
Λ
n
=
{
Z
=
n
}
, by referring to (1.4),
EE
[
X

Z
](
ω
) =
EE
[
X
; Λ
n
]IP(Λ
n
)=
EE
[
X
;
Z
=
n
]IP(
Z
=
n
)=
ϕ
(
n
) =
ϕ
(
Z
(
ω
))
.
(1
.
9)Equivalently,
EE
[
X

Z
](
ω
) =
ϕ
(
Z
)(
ω
) =
n
EE
[
X
;
Z
=
n
)IP(
Z
=
n
)
I
{
n
}
(
Z
(
ω
))
.
(1
.
10)This fact will be mentioned again in
§
2.4 below.7.
Remark.
Observe that
EE
[
X

Z
]
∈
σ
(
Z
)
,
i.e.,
EE
[
X

Z
] is a measurable w.r.t.
σ
(
Z
). By DoobDynkin Lemma (in
§
2.5 of Notes 1), we know that
EE
[
X

Z
] is now a
function
of
Z
, i.e., there is a Borelmeasurable function
ϕ
such that
EE
[
X

Z
] =
ϕ
(
Z
) a.s.4