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Christophe Profeta
Analíti a
k
7
Revista de Análisis Estadístico
Journal of Statistical Analysis
(5) and (6):
E
(
3
)
0
[
exp
(
−
λ
∫
+
∞
0
ds
(
R
2
s
+
2
aR
s
+
b
)
2
)]
=
E
(
3
)
0
[
exp
(
−
λ
∫
G
x
0
ds
(
R
2
s
+
2
aR
s
+
b
)
2
)]
E
(
3
)
0
exp
−
λ
∫
+
∞
0
ds
(
(
x
+
e
R
s
)
2
+
2
a
(
x
+
e
R
s
) +
b
)
2
=
E
x
[
exp
(
−
λ
∫
+
∞
0
M
2
s
(
M
2
s
+
2
a
M
s
+
b
)
2
ds
)]
E
(
3
)
0
[
exp
(
−
λ
∫
+
∞
0
ds
(
R
2
s
+ (
2
a
+
2
x
)
R
s
+
x
2
+
2
ax
+
b
)
2
)]
,
which yields the formula:
E
x
[
exp
(
−
λ
∫
+
∞
0
M
2
s
(
M
2
s
+
2
a
M
s
+
b
)
2
ds
)]
=
√
x
2
+
2
ax
+
b
√
b
sinh
(
√
2
λ
+
a
2
−
b
∫
+
∞
0
dz
z
2
+
2
(
a
+
x
)
z
+
x
2
+
2
ax
+
b
)
sinh
(
√
2
λ
+
a
2
−
b
∫
+
∞
0
dz
z
2
+
2
az
+
b
)
.
This new expression is seen to agree with Theorem 2 by
applying the change of variable
z
=
b
y
−
x
in the integral on
the numerator. The other relation follows as before thanks
to (4).
2.3 A few particular cases
i
)
When
a
=
1 and
b
=
1, we recover a particular case
of Hariya’s identity :
∫
+
∞
0
E
2
s
(
E
2
s
+
2
E
s
+
1
)
2
ds
=
∫
+
∞
0
ds
(
R
s
+
1
)
4
(law)
=
´ınf
{
t
≥
0,
R
t
=
1
}
.
with
E
0
=
x
and
R
0
=
1
−
1
1
+
x
.
ii
)
More generally, when
a
=
√
b
,
∫
+
∞
0
E
2
s
(
E
2
s
+
2
b
E
s
+
b
2
)
2
ds
=
∫
+
∞
0
ds
(
R
s
+
b
)
4
(law)
=
´ınf
{
t
≥
0,
R
t
=
1
b
}
,
with
E
0
=
x
and
R
0
=
1
b
−
1
x
+
b
.
iii
)
We may recover the result of Salminen-Yor [17] by
letting
b
→
0. Indeed, for 0
<
b
<
a
2
, we have:
∫
x
0
dz
z
2
+
2
az
+
b
=
1
2
√
a
2
−
b
(
ln
(
x
+
a
− √
a
2
−
b
x
+
a
+
√
a
2
−
b
)
−
ln
(
a
− √
a
2
−
b
a
+
√
a
2
−
b
))
so that, letting
b
go towards 0, we obtain:
sinh
( √
2
λ
+
a
2
−
b
∫
x
0
dz
z
2
+
2
az
+
b
)
∼
b
→
0
1
2
exp
(
√
2
λ
+
a
2
2
a
(
ln
(
x
x
+
2
a
)
−
ln
(
1
−
√
1
−
b
a
2
2
)))
and
E
x
[
exp
(
−
λ
∫
+
∞
0
ds
(
E
s
+
2
a
)
2
)]
=
√
x
+
2
a
√
x
(
x
x
+
2
a
)
√
2
λ
+
a
2
2
a
=
(
x
x
+
2
a
)
√
2
λ
+
a
2
2
a
−
1
2
.
This last expression is seen to coincide with the
Laplace transform of the first hitting time at level
1
2
a
ln
(
x
+
2
a
x
)
of a Brownian motion with drift
a
star-
ted from 0, which was the announced result in the
introduction, with
x
=
1.
We refer to Salminen & Yor [16, 17] and Decamps, De
Schepper, Goovaerts & Schoutens [4] for similar articles on
this subject.
3 Some Black-Scholes annuities
Let
(
B
t
+
ν
t
,
t
≥
0
)
be a standard Brownian motion
with drift
ν
started from 0. In this section, we study the
law of the pair of annuities :
(
∫
t
0
(
e
2
β
(
B
s
+
ν
s
)
−
1
)
+
ds
,
∫
t
0
(
e
2
b
(
B
s
+
ν
s
)
−
1
)
−
ds
)
(7)
where
x
+
=
m´ax
(
0,
x
)
and
x
−
=
m´ın
(
0,
x
)
.
T
HEOREM
5.
Let
α
,
β
,
a
,
b
,
λ
≥
0
and
ν
∈
R
. The double La-
place transform of the couple (7) is given by:
∫
+
∞
0
e
−
(
λ
+
a
)
t
E
0
[
exp
(
−
α
∫
t
0
(
e
2
β
(
B
s
+
ν
s
)
−
1
)
+
ds
−
a
∫
t
0
(
e
2
b
(
B
s
+
ν
s
)
−
1
)
−
ds
)]
dt
=
2
ω
λ
(
K
2
γ
(
√
2
α
β
)
∫
0
−
∞
e
ν
y
I
2
c
(
√
2
a
b
e
by
)
dy
+
I
2
c
(
√
2
a
b
)
∫
+
∞
0
e
ν
y
K
2
γ
(
√
2
α
β
e
β
y
)
dy
)
12
Analítika,
Revista de análisis estadístico
, 4 (2014), Vol. 7(1): 7-19