Versión de HTML Básico
Christophe Profeta
Analíti a
k
7
Revista de Análisis Estadístico
Journal of Statistical Analysis
1 Introduction
One of the most important concept in finance theory is
the
time value of money
which states that a given amount
of money at the present time is worth more that the same
amount in the future. This is due to the fact that you may
invest the money you hold today and earn interest. Assu-
ming that the rate of interest is constant and equal to
r
>
0,
the value of the money
V
(
k
)
at time
k
∈
N
satisfies:
V
(
k
) =
V
(
0
)
×
(
1
+
r
)
k
.
This formula may be used to compute the present value of
an annuity, that is, of a series of equal payments
P
that oc-
cur at evenly spaced intervals. For a period of
n
payments,
the present value of an annuity
A
n
(
0
)
equals:
A
n
(
0
) =
n
∑
k
=
1
P
(
1
+
r
)
k
=
P
r
(
1
−
1
(
1
+
r
)
n
)
.
(1)
Classic examples of annuities are for instance lease pay-
ments, insurance payments or regular deposits to a savings
account. Letting
n
tend towards
+
∞
in Formula (1), we ob-
tain the present value of a perpetuity
A
∞
(
0
)
, that is, when
the payments are not limited in time:
A
∞
(
0
) =
P
r
.
All these computations may be extended to a continuous-
time framework. In the continuous case, the value of the
money
V
(
s
)
at time
s
≥
0 satisfies :
V
(
s
) =
V
(
0
)
exp
(
rs
)
.
A natural generalization may be obtained by considering
varying interest rates, in which case
r
is no longer a positi-
ve constant but a function which fluctuates over time. The
value of the money
V
(
s
)
at time
s
≥
0 then satisfies :
V
(
s
) =
V
(
0
)
exp
(
∫
s
0
r
(
u
)
du
)
.
By analogy with (1), the present value of an annuity paid
continuously up to time
t
>
0 is thus given by :
A
t
(
0
) =
P
∫
t
0
exp
(
−
∫
s
0
r
(
u
)
du
)
ds
and, letting
t
tend to
+
∞
, the analogous perpetuity equals
:
A
∞
(
0
) =
P
∫
+
∞
0
exp
(
−
∫
s
0
r
(
u
)
du
)
ds
.
Unfortunately, the future evolution of interest rates (i.e. the
function
r
) is unknown, and is therefore often modeled by a
stochastic process. In other words, when dealing with long
term guaranteed payments, one is often led to study the
law of random variables of the form
∫
+
∞
0
X
s
ds
where
(
X
s
,
s
≥
0
)
is a given (positive) stochastic process.
One of the most famous example is certainly Dufresne’s
perpetuity. In his study of the value of a pension fund, Du-
fresne [5] proved the equality
∫
+
∞
0
e
aB
s
−
ν
s
ds
(law)
=
2
a
2
γ
2
ν
/
a
2
where
(
B
t
,
t
≥
0
)
is a Brownian motion started from 0 and
γ
µ
denotes a Gamma random variable with parameter
µ
:
P
(
γ
µ
∈
dz
) =
1
Γ
(
µ
)
e
−
z
z
µ
−
1
1
{
z
>
0
}
dz
.
Since then, many perpetuities involving geometric Brow-
nian motion
(
exp
(
ν
B
t
−
ν
2
t
2
)
,
t
≥
0
)
have been conside-
red, which is of no surprise due to its prominent role in the
celebrated Black-Scholes model. Observe that, in the set-up
of perpetuities, the parameter
ν
may be removed by scaling
since:
∫
+
∞
0
f
(
e
ν
B
s
±
ν
2
s
2
)
ds
(law)
=
∫
+
∞
0
f
(
e
B
ν
2
s
±
ν
2
s
2
)
ds
=
1
ν
2
∫
+
∞
0
f
(
e
B
u
±
u
2
)
du
.
Therefore, in the following, we shall emphasize our study
on the processes :
E
t
=
exp
(
B
t
+
t
2
)
and
M
t
=
exp
(
B
t
−
t
2
)
.
In [16], Salminen and Yor introduced a translated version
of Dufresne’s perpetuity, to circumvent the fact that the ori-
ginal one does not have all his moments finite. They prove
in particular the equality :
∫
+
∞
0
ds
(
2
a
+
E
s
)
2
(law)
=
´ınf
{
s
≥
0,
B
s
+
as
=
1
2
a
ln
(
1
+
2
a
)
}
which we shall recover in Section 2. In fact, many perpe-
tuities involving Brownian motion with drift are seen to be
identical in law with the first hitting time of some associa-
ted diffusions, see [17] for a discussion and many examples
on this subject.
In this paper, we study a generalization of Dufresne’s
translated perpetuity, that is,
∫
+
∞
0
E
2
s
(
E
2
s
+
2
a
E
s
+
b
)
2
ds
,
and show, in particular, the following equalities in law:
T
HEOREM
1.
Assume that b
>
0
and a
2
−
b
≥
0
. Let
(
R
(
3,
√
a
2
−
b
)
t
,
t
≥
0
)
be a three-dimensional Bessel process with
drift
√
a
2
−
b. Then, if
E
0
=
M
0
=
x
>
0
,
∫
+
∞
0
E
2
s
(
E
2
s
+
2
a
E
s
+
b
)
2
ds
(law)
=
´ınf
{
t
≥
0,
R
(
3,
√
a
2
−
b
)
t
=
η
(
∞
)
}
,
with R
(
3,
√
a
2
−
b
)
0
=
η
(
x
)
,
8
Analítika,
Revista de análisis estadístico
, 4 (2014), Vol. 7(1): 7-19