1The Lebesgue–Stieltjes integral
III Stochastic Calculus and Applications
1 The Lebesgue–Stieltjes integral
In calculus, we are able to perform integrals more exciting than simply
R
1
0
h
(
x
) d
x
.
In particular, if
h, a
: [0
,
1]
→ R
are
C
1
functions, we can perform integrals of
the form
Z
1
0
h(x) da(x).
For them, it is easy to make sense of what this means — it’s simply
Z
1
0
h(x) da =
Z
1
0
h(x)a
0
(x) dx.
In our world, we wouldn’t expect our functions to be differentiable, so this is
not a great definition. One reasonable strategy to make sense of this is to come
up with a measure that should equal “da”.
An immediate difficulty we encounter is that
a
0
(
x
) need not be positive all the
time. So for example,
R
1
0
1 d
a
could be a negative number, which one wouldn’t
expect for a usual measure! Thus, we are naturally lead to think about signed
measures.
From now on, we always use the Borel
σ
-algebra on [0
, T
] unless otherwise
specified.
Definition
(Signed measure)
.
A signed measure on [0
, T
] is a difference
µ
=
µ
+
−µ
−
of two positive measures on [0
, T
] of disjoint support. The decomposition
µ = µ
+
− µ
−
is called the Hahn decomposition.
In general, given two measures
µ
1
and
µ
2
with not necessarily disjoint
supports, we may still want to talk about µ
1
− µ
2
.
Theorem.
For any two finite measures
µ
1
, µ
2
, there is a signed measure
µ
with
µ(A) = µ
1
(A) −µ
2
(A).
If
µ
1
and
µ
2
are given by densities
f
1
, f
2
, then we can simply decompose
µ
as (
f
1
−f
2
)
+
d
t
+ (
f
1
−f
2
)
−
d
t
, where
+
and
−
denote the positive and negative
parts respectively. In general, they need not be given by densities with respect to
dx, but they are always given by densities with respect to some other measure.
Proof.
Let
ν
=
µ
1
+
µ
2
. By Radon–Nikodym, there are positive functions
f
1
, f
2
such that µ
i
(dt) = f
i
(t)ν(dt). Then
(µ
1
− µ
2
)(dt) = (f
1
− f
2
)
+
(t) · ν(dt) + (f
1
− f
2
)
−
(t) · ν(dt).
Definition
(Total variation)
.
The total variation of a signed measure
µ
=
µ
+
− µ
−
is |µ| = µ
+
+ µ
−
.
We now want to figure out how we can go from a function to a signed measure.
Let’s think about how one would attempt to define
R
1
0
f
(
x
) d
g
as a Riemann
sum. A natural option would be to write something like
Z
t
0
h(s) da(s) = lim
m→∞
n
m
X
i=1
h(t
(m)
i−1
)
a(t
(m)
i
) − a(t
(m)
i−1
)
for any sequence of subdivisions 0 =
t
(m)
0
< ··· < t
(m)
n
m
=
t
of [0
, t
] with
max
i
|t
(m)
i
− t
(m)
i−1
| → 0.
In particular, since we want the integral of
h
= 1 to be well-behaved, the
sum
P
(a(t
(m)
i
) − a(t
(m)
i−1
)) must be well-behaved. This leads to the notion of
Definition
(Total variation)
.
The total variation of a function
a
: [0
, T
]
→ R
is
V
a
(t) = |a(0)| + sup
(
n
X
i=1
|a(t
i
) − a(t
i−1
)| : 0 = t
0
< t
1
< ··· < t
n
= T
)
.
We say a has bounded variation if V
a
(T ) < ∞. In this case, we write a ∈ BV .
We include the
|a
(0)
|
term because we want to pretend
a
is defined on all of
R with a(t) = 0 for t < 0.
We also define
Definition
(C`adl`ag)
.
A function
a
: [0
, T
]
→ R
is c`adl`ag if it is right-continuous
and has left-limits.
The following theorem is then clear:
Theorem. There is a bijection
n
signed measures on [0, T ]
o
←→
c`adl`ag functions of bounded
variation a : [0, T ] → R
that sends a signed measure
µ
to
a
(
t
) =
µ
([0
, t
]). To construct the inverse, given
a, we define
a
±
=
1
2
(V
a
± a).
Then a
±
are both positive, and a = a
+
− a
−
. We can then define µ
±
by
µ
±
[0, t] = a
±
(t) − a
±
(0)
µ = µ
+
− µ
−
Moreover, V
µ[0,t]
= |µ|[0, t].
Example. Let a : [0, 1] → R be given by
a(t) =
(
1 t <
1
2
0 t ≥
1
2
.
This is c`adl`ag, and it’s total variation is
v
0
(1) = 2. The associated signed
measure is
µ = δ
0
− δ
1/2
,
and the total variation measure is
|µ| = δ
0
+ δ
1/2
.
We are now ready to define the Lebesgue–Stieltjes integral.
Definition
(Lebesgue–Stieltjes integral)
.
Let
a
: [0
, T
]
→ R
be c`adl`ag of
bounded variation and let
µ
be the associated signed measure. Then for
h ∈
L
1
([0, T ], |µ|), the Lebesgue–Stieltjes integral is defined by
Z
t
s
h(r) da(r) =
Z
(s,t]
h(r)µ(dr),
where 0 ≤ s ≤ t ≤ T , and
Z
t
s
h(r) |da(r)| =
Z
(s,t]
h(r)|µ|(dr).
We also write
h · a(t) =
Z
t
0
h(r) da(r).
To let T = ∞, we need the following notation:
Definition
(Finite variation)
.
A c`adl`ag function
a
: [0
, ∞
)
→ R
is of finite
variation if a|
[0,T ]
∈ BV [0, 1] for all T > 0.
Fact. Let a : [0, T ] → R be c`adl`ag and BV, and h ∈ L
1
([0, T ], |da|), then
Z
T
0
h(s) da(s)
≤
Z
b
a
|h(s)| |da(s)|,
and the function
h · a
: [0
, T
]
→ R
is c`adl`ag and BV with associated signed
measure h(s) da(s). Moreover, |h(s) da(s)| = |h(s)| |da(s)|.
We can, unsurprisingly, characterize the Lebesgue–Stieltjes integral by a
Riemann sum:
Proposition.
Let
a
be c`adl`ag and BV on [0
, t
], and
h
bounded and left-
continuous. Then
Z
t
0
h(s) da(s) = lim
m→∞
n
m
X
i=1
h(t
(m)
i−1
)
a(t
(m)
i
) − a(t
(m)
i−1
)
Z
t
0
h(s) |da(s)| = lim
m→∞
n
m
X
i=1
h(t
(m)
i−1
)
a(t
(m)
i
) − a(t
(m)
i−1
)
for any sequence of subdivisions 0 =
t
(m)
0
< ··· < t
(m)
n
m
=
t
of [0
, t
] with
max
i
|t
(m)
i
− t
(m)
i−1
| → 0.
Proof. We approximate h by h
m
defined by
h
m
(0) = 0, h
m
(s) = h(t
(m)
i−1
) for s ∈ (t
(m)
i−1
, t
(m)
i
].
Then by left continuity, we have
h(s) = lim
n→∞
h
m
(s)
by left continuity, and moreover
lim
m→∞
n
m
X
i=1
h(t
(m)
i−1
)(a(t
(m)
i
) − a(t
(m)
i−1
)) = lim
m→∞
Z
(0,t]
h
m
(s)µ( ds) =
Z
(0,t]
h(s)µ(ds)
by dominated convergence theorem. The statement about
|
d
a
(
s
)
|
is left as an
exercise.