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On Dufresne’s Translated Perpetuity and Some Black-Scholes Annuities
Analíti a
k
7
Revista de Análisis Estadístico
Journal of Statistical Analysis
=
2
|
ν
|
λ
K
2
γ
(
√
2
α
β
)
|
ν
|
K
2
γ
(
√
2
α
β
)
− √
2
α
K
′
2
γ
(
√
2
α
β
)
,
which agrees with [14, Section 4.4, p.107]. Unfortunately, it
does not seem easy to invert this Laplace transform.
4.2 The positive sojourn time of Brownian
motion with drift
Let first
α
→
0. From the asymptotics (see Section A.1):
K
ν
(
z
)
∼
z
→
0
2
ν
−
1
Γ
(
ν
)
z
ν
and
zK
′
ν
(
z
) =
−
zK
ν
−
1
(
z
)
−
ν
I
ν
(
z
)
∼
z
→
0
ν
2
ν
−
1
Γ
(
ν
)
z
ν
we deduce that
∫
+
∞
0
e
−
λ
t
E
0
[
e
−
at
−
a
∫
t
0
(
e
2
b
(
B s
+
ν
s
)
−
1
)
−
ds
]
dt
=
2
ω
λ
∫
0
−
∞
e
ν
y
I
2
γ
(
√
2
a
b
e
by
)
dy
+
I
2
c
(
√
2
a
b
)
√
2
(
λ
+
a
) +
ν
2
−
ν
where the Wronskien
ω
λ
equals:
ω
λ
=
√
2
aI
′
2
c
(
√
2
a
b
)
+
√
2
(
λ
+
a
) +
ν
2
I
2
c
(
√
2
a
b
)
.
We now let further
b
→
+
∞
. The left-hand side yields :
t
+
∫
t
0
(
e
2
b
(
B
s
+
ν
s
)
−
1
)
−
ds
=
t
+
∫
t
0
e
2
b
(
B
s
+
ν
s
)
1
{
B
s
+
ν
s
<
0
}
ds
−
∫
t
0
1
{
B
s
+
ν
s
<
0
}
ds
−−−−→
b
→
+
∞
∫
t
0
1
{
B
s
+
ν
s
≥
0
}
ds
To compute the limit as
b
→
+
∞
in the right-hand side, we
rely on the following integral formula
I
µ
(
z
) =
z
µ
2
µ
√
π
Γ
(
µ
+
1
2
)
∫
π
0
e
z
cos
(
θ
)
sin
2
µ
(
θ
)
d
θ
which gives :
I
√
2
λ
+
ν
2
b
(
√
2
a
b
e
by
)
−−−−→
b
→
+
∞
e
y
√
2
λ
+
ν
2
and
I
′√
2
λ
+
ν
2
b
(
√
2
a
b
)
=
I
√
2
λ
+
ν
2
b
+
1
(
√
2
a
b
)
+
√
2
λ
+
ν
2
√
2
a
I
√
2
λ
+
ν
2
b
(
√
2
a
b
)
−−−−→
b
→
+
∞
√
2
λ
+
ν
2
√
2
a
.
Therefore, we deduce that
∫
+
∞
0
e
−
λ
t
E
0
[
e
−
a
∫
t
0
1
{
B s
+
ν
s
>
0
}
ds
]
dt
=
2
ω
λ
(
∫
0
−
∞
e
ν
y
e
y
√
2
λ
+
ν
2
dy
+
1
√
2
(
λ
+
a
) +
ν
2
−
ν
)
=
2
ω
λ
(
1
ν
+
√
2
λ
+
ν
2
+
1
√
2
(
λ
+
a
) +
ν
2
−
ν
)
,
with
ω
λ
=
√
2
λ
+
ν
2
+
√
2
(
λ
+
a
) +
ν
2
.
We now study two further simplifications :
i
)
When
ν
=
0, this expression simplifies to:
∫
+
∞
0
e
−
λ
t
E
0
[
e
−
a
∫
t
0
1
{
B s
>
0
}
ds
]
dt
=
1
√
λ
√
λ
+
a
and this double Laplace transformmay be inverted to
recover the celebrated Arcsine law of Brownian mo-
tion:
P
0
(
∫
t
0
1
{
B
s
>
0
}
ds
∈
dz
)
=
1
π
√
z
√
t
−
z
1
{
0
<
z
<
t
}
dz
.
ii
)
When
ν
<
0, we may obtain the Laplace transform of
the associated perpetuity as before:
E
0
[
e
−
a
∫
+
∞
0
1
{
B s
+
ν
s
>
0
}
ds
]
=
l´ım
ε
→
0
2
λε
w
λε
(
1
ν
+
√
2
λε
+
ν
2
+
1
√
2
(
λε
+
a
) +
ν
2
−
ν
)
=
2
|
ν
|
|
ν
|
+
√
2
a
+
ν
2
,
which may be inverted to give (see [6, Formula 4,
p.233]):
P
0
(
∫
+
∞
0
1
{
B
s
+
ν
s
>
0
}
ds
∈
dz
)
=
|
ν
| √
2
√
π
(
1
√
z
e
−
ν
2
2
z
− |
ν
|
√
2
∫
+
∞
|
ν
| √
z
√
2
e
−
t
2
dt
)
1
{
z
>
0
}
dz
.
4.3 Yor’s functional
Take
a
=
α
and
b
=
β
. Then, from (11), the Wronskien
simplifies to
ω
λ
=
β
and from the formula (see [8, p.712]) :
2
I
γ
(
x
)
K
γ
(
y
) =
∫
+
∞
0
e
−
t
2
−
x
2
+
y
2
2
t
I
γ
(
xy
t
)
dt
t
,
for
y
≥
x
,
we obtain the expression :
∫
+
∞
0
e
−
λ
t
E
0
[
e
−
α
∫
t
0
e
2
β
(
B s
+
ν
s
)
ds
]
dt
=
1
β
∫
+
∞
0
dt
t
exp
(
−
t
2
−
α
t
β
2
)
∫
R
exp
(
−
α
t
β
2
e
2
β
y
)
Analítika,
Revista de análisis estadístico
, 4 (2014), Vol. 7(1): 7-19
15