Versión de HTML Básico
Christophe Profeta
Analíti a
k
7
Revista de Análisis Estadístico
Journal of Statistical Analysis
A Appendix on Bessel functions and
Bessel processes (with drift)
A.1 Modified Bessel functions [10, Chapter 5]
For
ν
∈
C
, let
I
ν
denote the modified Bessel function
defined by:
I
ν
(
x
) =
∞
∑
k
=
0
(
x
/2
)
ν
+
2
k
Γ
(
k
+
1
)
Γ
(
k
+
ν
+
1
)
x
>
0,
and
K
ν
the McDonald function defined, for
ν
/
∈
Z
, by:
K
ν
(
x
) =
π
2
I
−
ν
(
x
)
−
I
ν
(
x
)
sin
(
νπ
)
x
>
0,
and for
ν
=
n
∈
Z
by :
K
n
(
x
) =
l´ım
ν
→
n
ν
̸
=
n
K
ν
(
x
)
.
It is known that these functions generate the set of solu-
tions of the linear differential equation:
u
′′
+
1
x
u
′
−
(
1
+
ν
2
x
2
)
u
=
0.
Their derivatives are seen to satisfy several recurrence re-
lations:
xI
′
ν
(
x
) =
xI
ν
−
1
(
x
)
−
ν
I
ν
(
x
) =
xI
ν
+
1
(
x
) +
ν
I
ν
(
x
)
,
xK
′
ν
(
x
) =
−
xK
ν
−
1
(
x
)
−
ν
K
ν
(
x
) =
−
xK
ν
+
1
(
x
) +
ν
K
ν
(
x
)
,
and their Wronskien takes a particularly simple form:
W
(
I
ν
(
x
)
,
K
ν
(
x
))
:
=
I
ν
(
x
)
K
′
ν
(
x
)
−
I
′
ν
(
x
)
K
ν
(
x
) =
−
1
x
. (11)
Note that they both simplify when
ν
=
±
1/2:
I
−
1/2
(
z
) =
√
2
π
z
cosh
(
z
)
,
I
1/2
(
z
) =
√
2
π
z
sinh
(
z
)
and
K
−
1/2
(
z
) =
K
1/2
(
z
) =
√
π
2
z
e
−
z
.
For real and strictly positive
ν
>
0,
I
ν
is a positive increa-
sing function and
K
ν
is a positive decreasing function. In
this case, they both admit some useful integral representa-
tion:
I
ν
(
x
) =
x
ν
2
ν
√
π
Γ
(
ν
+
1
2
)
∫
π
0
e
x
cos
(
θ
)
sin
2
ν
(
θ
)
d
θ
K
ν
(
x
) =
1
2
(
x
2
)
ν
∫
+
∞
0
1
t
ν
+
1
exp
(
−
t
−
x
2
4
t
)
dt
from which we may deduce the following equivalents (still
for
ν
>
0):
I
ν
(
x
)
∼
x
→
0
x
ν
2
ν
Γ
(
ν
+
1
)
and
K
ν
(
x
)
∼
x
→
0
2
ν
−
1
Γ
(
ν
)
x
ν
,
and the asymptotic formulae when
x
→
+
∞
:
I
ν
(
x
)
∼
x
→
+
∞
e
x
√
2
π
x
and
K
ν
(
x
)
∼
x
→
+
∞
√
π
2
x
e
−
x
.
A.2 Bessel processes with drift [11]
Let
δ
∈
N
\{
0
}
and
c
>
0. The Bessel process of dimen-
sion
δ
(or equivalently of index
ν
=
δ
2
−
1) and drift
c
is the
diffusion
(
R
(
δ
,
c
)
t
,
t
≥
0
)
with generator:
G
=
1
2
∂
2
∂
x
2
+
(
2
ν
+
1
2
x
+
c
I
ν
+
1
I
ν
(
cx
)
)
∂
∂
x
.
(12)
We denote by
P
(
δ
,
c
)
x
its law when started from
x
. This
process may be obtained as the euclidean norm of a
δ
-
dimensional Brownian motion
−→
B
with drift
−→
µ
∈
R
δ
such
that
∥ −→
µ
∥
=
c
:
R
(
δ
,
c
)
t
=
∥ −→
B
t
+
−→
µ
−→
t
∥
.
The law of the first passage times of
(
R
t
,
t
≥
0
)
is given by
the following Laplace transform:
E
(
δ
,
c
)
x
[
e
−
λ
T
a
]
=
I
ν
(
x
√
2
λ
+
c
2
)
I
ν
(
a
√
2
λ
+
c
2
)
I
ν
(
ca
)
I
ν
(
cx
)
if
x
≤
a
,
K
ν
(
x
√
2
λ
+
c
2
)
K
ν
(
a
√
2
λ
+
c
2
)
I
ν
(
ca
)
I
ν
(
cx
)
if
x
≥
a
.
In particular, when
δ
=
3, (i.e.
ν
=
1
2
), these formulae sim-
plify to:
E
(
3,
c
)
x
[
e
−
λ
T
a
]
=
sinh
(
x
√
2
λ
+
c
2
)
sinh
(
a
√
2
λ
+
c
2
)
sinh
(
ca
)
sinh
(
cx
)
if
x
≤
a
,
exp
(
−
(
x
−
a
)
√
2
λ
+
c
2
)
sinh
(
ca
)
sinh
(
cx
)
if
x
≥
a
.
Note that this Laplace transform may be inverted, see [6,
p.258]:
18
Analítika,
Revista de análisis estadístico
, 4 (2014), Vol. 7(1): 7-19