2Semi-martingales
III Stochastic Calculus and Applications
2.1 Finite variation processes
The definition of a finite variation function extends immediately to a finite
variation process.
Definition
(Finite variation process)
.
A finite variation process is a c`adl`ag
adapted process
A
such that
A
(
ω, ·
) : [0
, ∞
)
→ R
has finite variation for all
ω ∈ Ω. The total variation process V of a finite variation process A is
V
t
=
Z
T
0
|dA
s
|.
Proposition.
The total variation process
V
of a c`adl`ag adapted process
A
is
also c`adl`ag, finite variation and adapted, and it is also increasing.
Proof.
We only have to check that it is adapted. But that follows directly
from our previous expression of the integral as the limit of a sum. Indeed, let
0 =
t
(m)
0
< t
(m)
1
< ··· < t
n
m
=
t
be a (nested) sequence of subdivisions of [0
, t
]
with max
i
|t
(m)
i
− t
(m)
i−1
| → 0. We have seen
V
t
= lim
m→∞
n
m
X
i=1
|A
t
(m)
i
− A
t
(m)
i−1
| + |A(0)| ∈ F
t
.
Definition
((
H ·A
)
t
)
.
Let
A
be a finite variation process and
H
a process such
that for all ω ∈ Ω and t ≥ 0,
Z
t
0
H
s
(ω)| |dA
s
(ω)| < ∞.
Then define a process ((H · A)
t
)
t≥0
by
(H ·A)
t
=
Z
t
0
H
s
dA
s
.
For the process H ·A to be adapted, we need a condition.
Definition
(Previsible process)
.
A process
H
: Ω
×
[0
, ∞
)
→ R
is previsible if
it is measurable with respect to the previsible
σ
-algebra
P
generated by the sets
E × (s, t], where E ∈ F
s
and s < t. We call the generating set Π.
Very roughly, the idea is that a previsible event is one where whenever it
happens, you know it a finite (though possibly arbitrarily small) before.
Definition
(Simple process)
.
A process
H
: Ω
×
[0
, ∞
)
→ R
is simple, written
H ∈ E, if
H(ω, t) =
n
X
i=1
H
i−1
(ω)1
(t
i−1
,t
i
]
(t)
for random variables H
i−1
∈ F
i−1
and 0 = t
0
< ··· < t
n
.
Fact. Simple processes and their limits are previsible.
Fact.
Let
X
be a c`adl`ag adapted process. Then
H
t
=
X
t
−
defines a left-
continuous process and is previsible.
In particular, continuous processes are previsible.
Proof.
Since
X
is c`adl`ag adapted, it is clear that
H
is left-continuous and
adapted. Since
H
is left-continuous, it is approximated by simple processes.
Indeed, let
H
n
t
=
2
n
X
i=1
H
(i−1)2
−n
1
((i−1)2
−n
,i2
−n
]
(t) ∧ n ∈ E.
Then H
n
t
→ H for all t by left continuity, and previsibility follows.
Exercise. Let H be previsible. Then
H
t
∈ F
t
−
= σ(F
s
: s < t).
Example. Brownian motion is previsible (since it is continuous).
Example. A Poisson process (N
t
) is not previsible since N
t
6∈ F
t
−
.
Proposition. Let A be a finite variation process, and H previsible such that
Z
t
0
|H(ω, s)| |dA(ω, s)| < ∞ for all (ω, t) ∈ Ω × [0, ∞).
Then H · A is a finite variation process.
Proof.
The finite variation and c`adl`ag parts follow directly from the deterministic
versions. We only have to check that
H · A
is adapted, i.e. (
H · A
)(
·, t
)
∈ F
t
for
all t ≥ 0.
First,
H ·A
is adapted if
H
(
ω, s
) = 1
(u,v]
(
s
)1
E
(
ω
) for some
u < v
and
E ∈ F
u
,
since
(H ·A)(ω, t) = 1
E
(ω)(A(ω, t ∧ v) − A(ω, t ∧ u)) ∈ F
t
.
Thus,
H ·A
is adapted for
H
=
1
F
when
F ∈
Π. Clearly, Π is a
π
system, i.e. it
is closed under intersections and non-empty, and by definition it generates the
previsible
σ
-algebra
P
. So to extend the adaptedness of
H · A
to all previsible
H, we use the monotone class theorem.
We let
V = {H : Ω × [0, ∞) → R : H · A is adapted}.
Then
(i) 1 ∈ V
(ii) 1
F
∈ V for all F ∈ Π.
(iii) V is closed under monotone limits.
So V contains all bounded P-measurable functions.
So the conclusion is that if
A
is a finite variation process, then as long as
reasonable finiteness conditions are satisfied, we can integrate functions against
d
A
. Moreover, this integral was easy to define, and it obeys all expected
properties such as dominated convergence, since ultimately, it is just an integral
in the usual measure-theoretic sense. This crucially depends on the fact that
A
is a finite variation process.
However, in our motivating example, we wanted to take
A
to be Brownian
motion, which is not of finite variation. The work we will do in this chapter
and the next is to come up with a stochastic integral where we let
A
be a
martingale instead. The heuristic idea is that while martingales can vary wildly,
the martingale property implies there will be some large cancellation between
the up and down movements, which leads to the possibility of a well-defined
stochastic integral.