Information Technology Reference
In-Depth Information
d
Definition 14.2.1
A function
F
:
S
→ R
,
S
⊂ R
is called
d
-
increasing
if
V
F
((a, b
]
)
≥
0 for all
a,b
∈
S
with
a
≤
b
and
(a, b
]⊂
S
.
Examples of
d
-increasing functions are distribution functions of random vectors
d
,
F(x
1
,...x
d
)
X
∈ R
= P[
X
1
≤
x
1
,...,X
d
≤
x
d
]
, or more generally,
=
−∞
,x
1
]
−∞
,x
d
]
F(x
1
,...x
d
)
μ ((
,...,(
) ,
d
)
.
F
is clearly
d
-increasing since the
F
-volume
where
μ
is a finite measure on
B
(
R
is just
V
F
((a, b
b
. For parametric modeling dependence
structures of multidimensional jump processes, margins play an important role.
]
)
=
μ ((a, b
]
)
for every
a
≤
d
Definition 14.2.2
Let
F
: R
→ R
be a
d
-increasing function which satisfies
F(u)
=
0if
u
i
=
0 for at least one
i
∈{
1
,...,
d
}
. For any index set
I
⊂{
1
,...,d
}
|
I
|
-margin of
F
is the function
F
I
: R
the
I
→ R
sgn
u
j
F(u
1
,...,u
d
).
F
I
(u
I
)
:=
lim
a
→∞
c
∈{−
a,
∞}
|
I
|
j
∈
I
c
(u
j
)
j
∈
I
c
d
)
, it is possible to define a suitable
notion of a copula. However, one has to take into account that the Lévy measure is
possibly infinite at the origin.
B
R
Since the Lévy measure is a measure on
(
d
Definition 14.2.3
A function
F
: R
→ R
is called a
Lévy copula
if
(i)
F(u
1
,...,u
d
)
= ∞
for
(u
1
,...,u
d
)
=
(
∞
,...,
∞
)
,
(ii)
F(u
1
,...,u
d
)
=
0if
u
i
=
0 for at least one
i
∈{
1
,...,d
}
,
(iii)
F
is
d
-increasing,
(iv)
F
{
i
}
(u)
=
u
for any
i
∈{
1
,...,d
}
,
u
∈ R
.
In the following, we prove some useful properties of Lévy copulas.
Lemma 14.2.4
Let F be a Lévy copula
.
Then
,
d
d
,
0
≤
sgn
u
j
F(u
1
,...,u
d
)
≤
min
{|
u
1
|
,...,
|
u
d
|}∀
u
∈ R
j
=
1
and
j
=
1
sgn
u
j
F(u)is nondecreasing in the absolute value of each argument
|
u
j
|
.
Furthermore
,
Lévy copulas are Lipschitz continuous
,
i
.
e
.
d
d
.
|
F(v
1
,...,v
d
)
−
F(u
1
,...,u
d
)
| ≤
1
|
v
i
−
u
i
|
∀
u, v
∈ R
i
=
d
,
u
1
≥
Proof
Let
u
∈ R
0 and 0
≤
a
1
≤
u
1
. Set
b
1
=
u
1
and for 2
≤
j
≤
d
set
a
j
=
0,
b
j
=
u
j
if
u
j
≥
0 otherwise
a
j
=
u
j
,
b
j
=
0. Since
F
is
d
-increasing and
grounded
Search WWH ::
Custom Search