2Semi-martingales
III Stochastic Calculus and Applications
2.3 Square integrable martingales
As previously discussed, we will want to use Hilbert space machinery to construct
the Itˆo integral. The rough idea is to define the Itˆo integral with respect to a
fixed martingale on simple processes via a (finite) Riemann sum, and then by
calculating appropriate bounds on how this affects the norm, we can extend this
to all processes by continuity, and this requires our space to be Hilbert. The
interesting spaces are defined as follows:
Definition (M
2
). Let
M
2
=
X : Ω × [0, ∞) → R : X is c´adl´ag martingale with sup
t≥0
E(X
2
t
) < ∞
.
M
2
c
=
X ∈ M
2
: X(ω, ·) is continuous for every ω ∈ Ω
We define an inner product on M
2
by
(X, Y )
M
2
= E(X
∞
Y
∞
),
which in aprticular induces a norm
kXk
M
2
=
E(X
2
∞
)
1/2
.
We will prove this is indeed an inner product soon. Here recall that for
X ∈ M
2
,
the martingale convergence theorem implies
X
t
→ X
∞
almost surely and in
L
2
.
Our goal will be to prove that these spaces are indeed Hilbert spaces. First
observe that if
X ∈ M
2
, then (
X
2
t
)
t≥0
is a submartingale by Jensen, so
t 7→ EX
2
t
is increasing, and
EX
2
∞
= sup
t≥0
EX
2
t
.
All the magic that lets us prove they are Hilbert spaces is Doob’s inequality.
Theorem (Doob’s inequality). Let X ∈ M
2
. Then
E
sup
t≥0
X
2
t
≤ 4E(X
2
∞
).
So once we control the limit
X
∞
, we control the whole path. This is why
the definition of the norm makes sense, and in particular we know
kXk
M
2
= 0
implies that X = 0.
Theorem. M
2
is a Hilbert space and M
2
c
is a closed subspace.
Proof.
We need to check that
M
2
is complete. Thus let (
X
n
)
⊆ M
2
be a Cauchy
sequence, i.e.
E((X
n
∞
− X
m
∞
)
2
) → 0 as n, m → ∞.
By passing to a subsequence, we may assume that
E((X
n
∞
− X
n−1
∞
)
2
) ≤ 2
−n
.
First note that
E
X
n
sup
t≥0
|X
n
t
− X
n−1
t
|
!
≤
X
n
E
sup
t≥0
|X
n
t
− X
n−1
t
|
2
1/2
(CS)
≤
X
n
2E
|X
n
∞
− X
n−1
∞
|
2
1/2
(Doob’s)
≤ 2
X
n
2
−n/2
< ∞.
So
∞
X
n=1
sup
t≥0
|X
n
t
− X
n−1
t
| < ∞ a.s. (∗)
So on this event, (
X
n
) is a Cauchy sequence in the space (
D
[0
, ∞
)
, k · k
∞
) of
c´adl´ag sequences. So there is some X(ω, ·) ∈ D[0, ∞) such that
kX
n
(ω, ·) − X(ω, ·)k
∞
→ 0 for almost all ω.
and we set X = 0 outside this almost sure event (∗). We now claim that
E
sup
t≥0
|X
n
− X|
2
→ 0 as n → ∞.
We can just compute
E
sup
t
|X
n
− X|
2
= E
lim
m→∞
sup
t
|X
n
− X
m
|
2
≤ lim inf
m→∞
E
sup
t
|X
n
− X
m
|
2
(Fatou)
≤ lim inf
m→∞
4E(X
n
∞
− X
∞
m
)
2
(Doob’s)
and this goes to 0 in the limit n → ∞ as well.
We finally have to check that
X
is indeed a martingale. We use the triangle
inequality to write
kE(X
t
| F
s
) − X
s
k
L
2
≤ kE(X
t
− X
n
t
| F
s
)k
L
2
+ kX
n
s
− X
s
k
L
2
≤ E(E((X
t
− X
n
t
)
2
| F
s
))
1/2
+ kX
n
s
− X
s
k
L
2
= kX
t
− X
n
t
k
L
2
+ kX
n
s
− X
s
k
L
2
≤ 2E
sup
t
|X
t
− X
n
t
|
2
1/2
→ 0
as
n → ∞
. But the left-hand side does not depend on
n
. So it must vanish. So
X ∈ M
2
.
We could have done exactly the same with continuous martingales, so the
second part follows.