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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
where the Wronskien
ω
λ
equals:
ω
λ
=
√
2
aK
2
γ
(
√
2
α
β
)
I
′
2
c
(
√
2
a
b
)
−
√
2
α
K
′
2
γ
(
√
2
α
β
)
I
2
c
(
√
2
a
b
)
and I
2
c
and K
2
γ
denote the third modified Bessel functions with
respective indexes
c
=
1
2
b
√
2
λ
+
ν
2
and
γ
=
1
2
β
√
2
(
λ
+
a
)
−
2
α
+
ν
2
if
2
(
λ
+
a
)
−
2
α
+
ν
2
≥
0
i
2
β
√
2
α
−
2
(
λ
+
a
)
−
ν
2
otherwise.
The proof of this result is given in the next Sections 3.1
and 3.2. We will discuss in Section 4 some special cases for
which the expression on the right-hand side simplifies.
3.1 A useful version of the Feynman-Kac for-
mula
To prove Theorem 5, we shall apply the following well-
known Feynman-Kac formula, see for instance Janson [9,
Appendix C] where many other Brownian areas are also
studied.
T
HEOREM
6
(Feynman-Kac)
.
Let V
(
x
)
≥
0
be a positive con-
tinuous function on
R
,
λ
>
0
, and let
ϕ
+
and
ϕ
−
be two
C
2
-
solutions of the differential equation
1
2
ϕ
′′
(
x
) = (
V
(
x
) +
λ
)
ϕ
(
x
)
(8)
such that, for A large enough :
ϕ
+
is positive and bounded on
[
A
,
+
∞
[
and
ϕ
−
is positive and bounded on
]
−
∞
,
−
A
]
.
(9)
Let w
λ
:
=
ϕ
+
(
0
)
ϕ
′ −
(
0
)
−
ϕ
−
(
0
)
ϕ
′
+
(
0
)
and assume that
ω
λ
̸
=
0
. Then, for any positive and measurable function f on
R
and
any x
∈
R
:
∫
+
∞
0
e
−
λ
t
E
x
[
e
−
∫
t
0
V
(
B
s
)
ds
f
(
B
t
)
]
dt
=
2
ω
λ
(
ϕ
+
(
x
)
∫
x
−
∞
ϕ
−
(
y
)
f
(
y
)
dy
+
ϕ
−
(
x
)
∫
+
∞
x
ϕ
+
(
y
)
f
(
y
)
dy
)
.
(10)
Demostración.
We sketch the proof of this result for the sake of complete-
ness. First, define the Wronskien:
W
λ
(
x
) =
ϕ
+
(
x
)
ϕ
′ −
(
x
)
−
ϕ
−
(
x
)
ϕ
′
+
(
x
)
.
Since
ϕ
+
and
ϕ
−
are solutions of (8), we deduce that
W
′
λ
(
x
) =
0, hence for any
x
∈
R
,
W
λ
(
x
) =
W
λ
(
0
) =
ω
λ
.
Assume first that
f
is continuous and has compact support,
and define:
ϕ
(
x
) =
ϕ
+
(
x
)
∫
x
−
∞
ϕ
−
(
y
)
f
(
y
)
dy
+
ϕ
−
(
x
)
∫
+
∞
x
ϕ
+
(
y
)
f
(
y
)
dy
.
ϕ
is a function of
C
1
-class, and differentiation yields:
ϕ
′
(
x
) =
ϕ
′
+
(
x
)
∫
x
−
∞
ϕ
−
(
y
)
f
(
y
)
dy
+
ϕ
′ −
(
x
)
∫
+
∞
x
ϕ
+
(
y
)
f
(
y
)
dy
.
We thus deduce that
ϕ
is of
C
2
-class, and from (8):
ϕ
′′
(
x
) =
2
(
V
(
x
) +
λ
)
ϕ
(
x
)
−
W
λ
(
x
)
f
(
x
)
=
2
(
V
(
x
) +
λ
)
ϕ
(
x
)
−
ω
λ
f
(
x
)
.
Observe also that, since
f
is a function with compact sup-
port, the function
ϕ
is bounded on
R
. Consider now the
process
M
t
=
e
−
λ
t
−
∫
t
0
V
(
B
s
)
ds
ϕ
(
B
t
) +
ω
λ
2
∫
t
0
e
−
λ
u
−
∫
u
0
V
(
B
s
)
ds
f
(
B
u
)
du
.
From Itô’s formula, this process is a local martingale and
we have the estimate:
|
M
t
| ≤
sup
x
∈
R
|
ϕ
(
x
)
|
+
ω
λ
2
sup
x
∈
R
|
f
(
x
)
|
∫
t
0
e
−
λ
u
du
≤
sup
x
∈
R
|
ϕ
(
x
)
|
+
ω
λ
2
λ
sup
x
∈
R
|
f
(
x
)
|
.
Therefore
M
is uniformly bounded, i.e.
M
is a bounded
martingale and
ϕ
(
x
) =
E
x
[
M
0
] =
E
x
[
M
∞
]
=
ω
λ
2
E
x
[
∫
+
∞
0
e
−
λ
u
−
∫
u
0
V
(
B
s
)
ds
f
(
B
u
)
du
]
.
By a monotone class argument, the assumption on the con-
tinuity of
f
may be dropped, so Relation (10) is in fact va-
lid for any positive and measurable function with compact
support. Let now
f
by a positive and measurable function,
and consider the sequence of functions
f
n
(
y
) =
f
(
y
)
1
{|
y
|≤
n
}
.
Since the
f
n
have compact support, we may apply Relation
(10) and write, for
n
large enough:
∫
+
∞
0
e
−
λ
t
E
x
[
e
−
∫
t
0
V
(
B
s
)
ds
f
(
B
t
)
1
{|
B
t
|≤
n
}
]
dt
=
2
ω
λ
(
ϕ
+
(
x
)
∫
x
−
n
ϕ
−
(
y
)
f
(
y
)
dy
+
ϕ
−
(
x
)
∫
n
x
ϕ
+
(
y
)
f
(
y
)
dy
)
.
We finally end the proof by letting
n
→
+
∞
and applying
the monotone convergence theorem thanks to the Condi-
tion (9).
Analítika,
Revista de análisis estadístico
, 4 (2014), Vol. 7(1): 7-19
13