2Semi-martingales
III Stochastic Calculus and Applications
2.5 Covariation
We know
M
2
c
not only has a norm, but also an inner product. This can also be
reflected in the bracket by the polarization identity, and it is natural to define
Definition
(Covariation)
.
Let
M, N
be two continuous local martingales. Define
the covariation (or simply the bracket) between M and N to be process
hM, Ni
t
=
1
4
(hM + Ni
t
− hM − Ni
t
).
Then if in fact M, N ∈ M
2
c
, then putting t = ∞ gives the inner product.
Proposition.
(i) hM, Ni
is the unique (up to indistinguishability) finite variation process
such that M
t
N
t
− hM, Ni
t
is a continuous local martingale.
(ii) The mapping (M, N ) 7→ hM, N i is bilinear and symmetric.
(iii)
hM, Ni
t
= lim
n→∞
hM, Ni
(n)
t
u.c.p.
hM, Ni
(n)
t
=
d2
n
te
X
i=1
(M
i2
−n
−M
(i−1)2
−n
)(N
i2
−n
− N
(i−1)
2
−n
).
(iv) For every stopping time T ,
hM
T
, N
T
i
t
= hM
T
, Ni
t
= hM, Ni
t∧T
.
(v)
If
M, N ∈ M
2
c
, then
M
t
N
t
−hM, Ni
t
is a uniformly integrable martingale,
and
hM − M
0
, N − N
0
i
M
2
= EhM, Ni
∞
.
Example.
Let
B, B
0
be two independent Brownian motions (with respect to
the same filtration). Then hB, B
0
i = 0.
Proof.
Assume
B
0
=
B
0
0
= 0. Then
X
±
=
1
√
2
(
B ± B
0
) are Brownian motions,
and so hX
±
i = t. So their difference vanishes.
An important result about the covariation is the following Cauchy–Schwarz
like inequality:
Proposition
(Kunita–Watanabe)
.
Let
M, N
be continuous local martingales
and let H, K be two (previsible) processes. Then almost surely
Z
∞
0
|H
s
||K
s
||dhM, Ni
s
| ≤
Z
∞
0
H
2
s
dhMi
s
1/2
Z
∞
0
H
2
s
hNi
s
1/2
.
In fact, this is Cauchy–Schwarz. All we have to do is to take approximations
and take limits and make sure everything works out well.
Proof. For convenience, we write
hM, Ni
t
s
= hM, Ni
t
− hM, N i
s
.
Claim. For all 0 ≤ s ≤ t, we have
|hM, Ni
t
s
| ≤
p
hM, Mi
t
s
p
hN, Ni
t
s
.
By continuity, we can assume that s, t are dyadic rationals. Then
|hM, Ni
t
s
| = lim
n→∞
2
n
t
X
i=2
n
s+1
(M
i2
−n
− M
(i−1)2
−n
)(N
i2
−n
− N
(i−1)2
−n
)
≤ lim
n→∞
2
n
t
X
i=2
n
s+1
(M
i2
−n
− M
(i−1)2
−n
)
2
1/2
×
2
n
t
X
i=2
n
s+1
(N
i2
−n
− N
(i−1)2
−n
)
2
1/2
(Cauchy–Schwarz)
=
hM, Mi
t
s
1/2
hN, Ni
t
s
1/2
,
where all equalities are u.c.p.
Claim. For all 0 ≤ s < t, we have
Z
t
s
|dhM, Ni
u
| ≤
p
hM, Mi
t
s
p
hN, Ni
t
s
.
Indeed, for any subdivision s = t
0
< t
1
< ···t
n
= t, we have
n
X
i=1
|hM, Ni
t
i
t
i−1
| ≤
n
X
i=1
q
hM, Mi
t
i
t
i−1
q
hN, Ni
t
i
t
i−1
≤
n
X
i=1
hM, Mi
t
i
t
i−1
!
1/2
n
X
i=1
hN, Ni
t
i
t
i−1
!
1/2
.
(Cauchy–Schwarz)
Taking the supremum over all subdivisions, the claim follows.
Claim. For all bounded Borel sets B ⊆ [0, ∞), we have
Z
B
|dhM, Ni
u
| ≤
s
Z
B
dhMi
u
s
Z
B
dhNi
u
.
We already know this is true if
B
is an interval. If
B
is a finite union of
integrals, then we apply Cauchy–Schwarz. By a monotone class argument, we
can extend to all Borel sets.
Claim. The theorem holds for
H =
k
X
`=1
h
`
1
B
`
, K =
n
X
`=1
k
`
1
B
`
for B
`
⊆ [0, ∞) bounded Borel sets with disjoint support.
We have
Z
|H
s
K
s
| |dhM, N i
s
| ≤
n
X
`=1
|h
`
k
`
|
Z
B
`
|d
¯
M, Ni
s
|
≤
n
X
`=1
|h
`
k
`
|
Z
B
`
dhMi
s
1/2
Z
B
`
dhNi
s
1/2
≤
n
X
`=1
h
2
`
Z
B
`
dhMi
s
!
1/2
n
X
`=1
k
2
`
Z
B
`
dhNi
s
!
1/2
To finish the proof, approximate general
H
and
K
by step functions and take
the limit.