The intensity
λ
of a counting process is a measure of the rate of change of its predictable part. If a stochastic process
{
N
(
t
)
,
t
≥
0
}
is a counting process, then it is a submartingale, and in particular its Doob-Meyer decomposition is
N
(
t
)
=
M
(
t
)
+
Λ
(
t
)
where
M
(
t
)
is a martingale and
Λ
(
t
)
is a predictable increasing process.
Λ
(
t
)
is called the cumulative intensity of
N
(
t
)
and it is related to
λ
by
Λ
(
t
)
=
∫
0
t
λ
(
s
)
d
s
.
Given probability space
(
Ω
,
F
,
P
)
and a counting process
{
N
(
t
)
,
t
≥
0
}
which is adapted to the filtration
{
F
t
,
t
≥
0
}
, the intensity of
N
is the process
{
λ
(
t
)
,
t
≥
0
}
defined by the following limit:
λ
(
t
)
=
lim
h
↓
0
1
h
E
[
N
(
t
+
h
)
−
N
(
t
)
|
F
t
]
.
The right-continuity property of counting processes allows us to take this limit from the right.
In statistical learning, the variation between
λ
and its estimator
λ
^
can be bounded with the use of oracle inequalities.
If a counting process
N
(
t
)
is restricted to
t
∈
[
0
,
1
]
and
n
i.i.d. copies are observed on that interval,
N
1
,
N
2
,
…
,
N
n
, then the least squares functional for the intensity is
R
n
(
λ
)
=
∫
0
1
λ
(
t
)
2
d
t
−
2
n
∑
i
=
1
n
∫
0
1
λ
(
t
)
d
N
i
(
t
)
which involves an Ito integral. If the assumption is made that
λ
(
t
)
is piecewise constant on
[
0
,
1
]
, i.e. it depends on a vector of constants
β
=
(
β
1
,
β
2
,
…
,
β
m
)
∈
R
+
m
and can be written
λ
β
=
∑
j
=
1
m
β
j
λ
j
,
m
,
λ
j
,
m
=
m
1
(
j
−
1
m
,
j
m
]
,
where the
λ
j
,
m
have a factor of
m
so that they are orthonormal under the standard
L
2
norm, then by choosing appropriate data-driven weights
w
^
j
which depend on a parameter
x
>
0
and introducing the weighted norm
∥
β
∥
w
^
=
∑
j
=
2
m
w
^
j
|
β
j
−
β
j
−
1
|
,
the estimator for
β
can be given:
β
^
=
arg
min
β
∈
R
+
m
{
R
n
(
λ
β
)
+
∥
β
∥
w
^
}
.
Then, the estimator
λ
^
is just
λ
β
^
. With these preliminaries, an oracle inequality bounding the
L
2
norm
∥
λ
^
−
λ
∥
is as follows: for appropriate choice of
w
^
j
(
x
)
,
∥
λ
^
−
λ
∥
2
≤
inf
β
∈
R
+
m
{
∥
λ
β
−
λ
∥
2
+
2
∥
β
∥
w
^
}
with probability greater than or equal to
1
−
12.85
e
−
x
.
