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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
P
(
3,
c
)
x
(
T
a
∈
dt
)
/
dt
=
sinh
(
ca
)
sinh
(
cx
)
1
√
2
π
t
3
+
∞
∑
n
=
−
∞
((
2
n
+
1
)
a
−
x
)
exp
(
−
((
2
n
+
1
)
a
−
x
)
2
2
t
−
c
2
2
t
)
if
x
≤
a
,
sinh
(
ca
)
sinh
(
cx
)
(
x
−
a
)
√
2
π
t
3
exp
(
−
(
x
−
a
)
2
2
t
−
c
2
2
t
)
if
x
≥
a
.
A.3 Bessel processes
Letting
c
→
0 informally in (12), we obtain the genera-
tor of the classic Bessel process with index
ν
:
G
=
1
2
∂
2
∂
x
2
+
2
ν
+
1
2
x
∂
∂
x
.
(13)
These processes are deeply related with geometric Brow-
nian motion thanks to the following Lamperti’s relation:
T
HEOREM
7
([18])
.
Let
(
B
t
+
ν
t
,
t
≥
0
)
be a Brownian mo-
tion with drift
ν
>
0
starting from
log
(
x
)
. Then, there exists
(
R
(
ν
)
t
,
t
≥
0
)
a Bessel process of index
ν
started from x such
that:
exp
(
B
t
+
ν
t
) =
R
(
ν
)
∫
t
0
exp
(
2
(
B
s
+
ν
s
))
ds
.
In particular, for
ν
>
0,
(
R
(
ν
)
t
,
t
≥
0
)
is transient and
l´ım
t
→
+
∞
R
(
ν
)
t
= +
∞
a.s. In the framework of perpetuities, we
may mention the following theorem, which allows to com-
pute the law of some perpetuities involving squared Bessel
processes, see [15, Theorem 1.7 p.444]:
T
HEOREM
8.
Let
φ
be a positive and measurable function such
that
∫
+
∞
0
(
1
+
x
)
φ
(
x
)
dx
<
+
∞
. Then:
E
(
δ
)
x
[
exp
(
−
∫
+
∞
0
R
2
t
φ
(
t
)
dt
)]
=
F
(+
∞
)
δ
/2
exp
(
x
2
F
′
(
0
)
)
where F is the unique solution on
[
0,
+
∞
[
of :
F
′′
=
φ
(
x
)
F such that F is positive, non increasing, and
F
(
0
) =
1
.
A.4 The three-dimensional Bessel process [15,
Chapter VI.3]
We now take
δ
=
3 (i.e.
ν
=
1
2
) in (13) to obtain the
classic three-dimensional Bessel process. This process en-
joys many important properties and is, in some sense, very
close de Brownian motion. In particular, there is a weak ab-
solute continuity formula between Brownian motion and
the three-dimensional Bessel process :
P
(
3
)
x
|F
t
=
B
t
x
1
{
t
>
T
0
}
P
x
|F
t
.
(14)
The paths of a three-dimensional Bessel process admit a
useful decomposition as follows:
T
HEOREM
9.
Let
(
R
t
,
t
≥
0
)
be a three-dimensional Bessel pro-
cess started from 0 and define G
x
=
sup
{
t
≥
0,
R
t
=
x
}
it last
passage time at level x. Then:
i
)
conditionally on G
x
, the processes
(
R
t
,
t
≤
G
x
)
and
(
R
t
+
G
x
,
t
≥
0
)
are independent,
ii
)
the process
(
R
t
+
G
x
,
t
≥
0
)
has the same law as
(
x
+
R
t
,
t
≥
0
)
,
iii
)
the process
(
R
t
,
t
≤
G
x
)
has the same law as
(
B
T
0
−
t
,
t
≤
T
0
)
where
(
B
t
,
t
≥
0
)
is a Brownian motion started from
x and T
0
=
´ınf
{
t
≥
0,
B
t
=
0
}
.
This result was first proven by Williams [18] in its decom-
position of the Brownian paths, see also Pitman [12] for a
related study.
We conclude this appendix by stating the third Ray-Knight
theorem, which describes the dependence in the space va-
riable of the total local time of the three-dimensional Bessel
process :
T
HEOREM
10
(Ray-Knight)
.
Let
(
R
t
,
t
≥
0
)
be a three-
dimensional Bessel process started from 0 and denote by L
y
∞
(
R
)
its total local time at level y. Then:
(
L
y
∞
,
y
≥
0
)
(law)
= (
Z
(
2
)
y
,
y
≥
0
)
where Z
(
2
)
is a two-dimensional squared Bessel process started
from 0.
We refer to [2, Chapter V] for other similar Ray-Knight
theorems.
Analítika,
Revista de análisis estadístico
, 4 (2014), Vol. 7(1): 7-19
19