3The stochastic integral
III Stochastic Calculus and Applications
3.3 Extension to local martingales
We have now defined the stochastic integral for continuous martingales. We next
go through some formalities to extend this to local martingales, and ultimately
to semi-martingales. We are not doing this just for fun. Rather, when we
later prove results like Itˆo’s formula, even when we put in continuous (local)
martingales, we usually end up with some semi-martingales. So it is useful to be
able to deal with semi-martingales in general.
Definition (L
2
bc
(M)). Let L
2
bc
(M) be the space of previsible H such that
Z
t
0
H
2
s
dhMi
s
< ∞ a.s.
for all finite t > 0.
Theorem. Let M be a continuous local martingale.
(i)
For every
H ∈ L
2
bc
(
M
), there is a unique continuous local martingale
H ·M
with (H · M)
0
= 0 and
hH · M, Ni = H · hM, Ni
for all N, M .
(ii) If T is a stopping time, then
(1
[0,T ]
H) ·M = (H · M)
T
= H · M
T
.
(iii)
If
H ∈ L
2
loc
(
M
),
K
is previsible, then
K ∈ L
2
loc
(
H ·M
) iff
HK ∈ L
2
loc
(
M
),
and then
K · (H · M) = (KH) ·M.
(iv)
Finally, if
M ∈ M
2
c
and
H ∈ L
2
(
M
), then the definition is the same as
the previous one.
Proof.
Assume
M
0
= 0, and that
R
t
0
H
2
s
d
hMi
s
< ∞
for all
ω ∈
Ω (by setting
H = 0 when this fails). Set
S
n
= inf
t ≥ 0 :
Z
t
0
(1 + H
2
s
) dhMi
s
≥ n
.
These S
n
are stopping times that tend to infinity. Then
hM
S
n
, M
S
n
i
t
= hM, M i
t∧S
n
≤ n.
So M
S
n
∈ M
2
c
. Also,
Z
∞
0
H
s
dhM
S
n
i
s
=
Z
S
n
0
H
2
s
dhMi
s
≤ n.
So
H ∈ L
2
(
M
S
n
), and we have already defined what
H · M
S
n
is. Now notice
that
H · M
S
n
= (H · M
S
m
)
S
n
for m ≥ n.
So it makes sense to define
H · M = lim
n→∞
H · M
S
n
.
This is the unique process such that (
H · M
)
S
n
=
H · M
S
n
. We see that
H · M
is a continuous adapted local martingale with reducing sequence S
n
.
Claim. hH · M, N i = H · hM, Ni.
Indeed, assume that
N
0
= 0. Set
S
0
n
=
inf{t ≥
0 :
|N
t
| ≥ n}
. Set
T
n
=
S
n
∧ S
0
n
. Observe that N
S
0
n
∈ M
2
c
. Then
hH · M, Ni
T
n
= hH · M
S
n
, N
S
0
n
i = H · hM
S
n
, N
S
0
n
i = H · hM, Ni
T
n
.
Taking the limit n → ∞ gives the desired result.
The proofs of the other claims are the same as before, since they only use
the characterizing property hH · M, Ni = H · hM, Ni.