Versión de HTML Básico
Christophe Profeta
Analíti a
k
7
Revista de Análisis Estadístico
Journal of Statistical Analysis
e
ν
y
I
2
γ
(
2
α
e
β
y
t
β
2
)
dy
=
1
β
∫
+
∞
0
dz
z
e
−
α
z
exp
(
−
1
2
z
β
2
)
∫
R
exp
(
−
1
2
z
β
2
e
2
β
y
)
e
ν
y
I
2
γ
(
e
β
y
z
β
2
)
dy
.
We may therefore invert the Laplace transform in
α
to ob-
tain:
P
0
(
∫
τ
λ
0
e
2
β
(
B
s
+
ν
s
)
ds
∈
dz
)
=
λ
z
β
e
−
1
2
z
β
2
∫
R
exp
(
−
1
2
z
β
2
e
2
β
y
)
e
ν
y
I
√
2
λ
+
ν
2
β
(
e
β
y
z
β
2
)
dy
where
τ
λ
denotes an exponential random variable with pa-
rameter
λ
independent from
B
.
Note that this last Laplace transform may be also inverted
thanks to the Hartman-Watson function
θ
r
given by:
I
√
2
λ
(
r
) =
∫
+
∞
0
e
−
λ
t
θ
r
(
t
)
dt
which, from Yor [19], admits the representation:
θ
r
(
t
) =
r
√
2
π
3
t
∫
+
∞
0
e
π
2
−
y
2
2
t
−
r
cosh
(
y
)
sinh
(
y
)
sin
(
π
y
t
)
dy
.
4.4 One-sided Yor’s functional
Take
a
=
α
and let
b
→
+
∞
. We obtain :
∫
+
∞
0
e
−
λ
t
E
0
[
e
−
α
∫
t
0
e
2
β
(
B s
+
ν
s
)
1
{
B s
+
ν
s
>
0
}
ds
]
dt
=
2
ω
λ
K
2
γ
(
√
2
α
β
)
ν
+
√
2
λ
+
ν
2
+
∫
+
∞
0
e
ν
y
K
2
γ
(
√
2
α
β
e
β
y
)
with
ω
λ
=
K
2
γ
(
√
2
α
β
) √
2
λ
+
ν
2
−
√
2
α
K
′
2
γ
(
√
2
α
β
)
.
This allows to recover the associated perpetuity, for
ν
<
0 :
E
0
[
exp
(
−
α
∫
+
∞
0
e
2
β
(
B
s
+
ν
s
)
1
{
B
s
+
ν
s
>
0
}
ds
)]
=
2
|
ν
|
K
|
ν
|
β
(
√
2
α
β
)
√
2
α
K
|
ν
|
β
+
1
(
√
2
α
β
)
.
We refer to Salminen & Yor [16, 17] for a comprehensive
study of this family of perpetuities.
Referencias
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BRAMOWITZ AND
I. A. S
TEGUN
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16
Analítika,
Revista de análisis estadístico
, 4 (2014), Vol. 7(1): 7-19