Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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660 A List of Special Features, Probability Laws, and Functions
NIG(α, β, μ, δ), |β| <α,δ>0
Density a
q
x − μ
δ
−1
K
1
δαq
x − μ
δ
e
β(x−μ)
q(x)=
√
1+x
2
a = π
−1
α exp
δ
α
2
− β
2
Characteristic f. exp
−δ
α
2
− (β + iu)
2
−
α
2
− β
2
− iuμ
Mean
δβ
α
2
− β
2
+ μ
Variance α
2
δ(α
2
− β
2
)
3/2
L´evy density
δα
π
e
βx
K
1
(α|x|)
|x|
In particular IG(a, ν)=GIG(−
1
2
,a,ν).
A.4 Some Important Probability Distributions 661
A.4.6 Variance Gamma VG(σ, ν, θ)
Density c|x|
1
ν
−
1
2
e
θ/σ
2
K
1
ν
−
1
2
√
θ
2
+2σ
2
/ν
σ
2
|x|
c = σ
ν
2π
(θ
2
ν+2σ
2
)
1
4
−
θ
2ν
Γ (1/ν)
VG(σ, ν, θ) Characteristic function
1 − iuθν +
1
2
σ
2
νu
2
−1/ν
Mean θ
Variance σ
2
+ νθ
2
L´evy density
1
ν|x|
e
Ax−B|x|
A =
θ
σ
2
,B=
√
θ
2
+2σ
2
/ν
σ
2
A.4.7 Tempered Stable TS(Y
±
,C
±
,M
±
)
L´evy density ν(x)=
C
+
x
Y
+
+1
e
−M
+
x
1
x>0
+
C
−
|x|
Y
−
+1
e
M
−
x
1
x<0
Characteristic function Γ (−Y
+
)M
Y
+
+
C
+
(1 −
iu
M
+
)
Y
+
− 1+
iuY
+
M
+
(for Y
±
=1, 0) +Γ (−Y
−
)M
Y
−
−
C
−
(1 +
iu
M
−
)
Y
−
− 1 −
iuY
−
M
−
Mean 0
Variance Γ(2 − Y
+
)C
+
M
Y
+
−2
+
+Γ (2 − Y
−
)C
−
M
Y
−
−2
−
662 A List of Special Features, Probability Laws, and Functions
A.5 Special Functions
A.5.1 Gamma and Beta Functions
Let a, b ∈ R
+
Γ (a)=
+∞
0
x
a−1
exp(−x)dx ,
B(a, b)=
Γ (a)Γ (b)
Γ (a + b)
A.5.2 Bessel Functions
The Bessel functions of the first kind J
ν
and of the second kind N
ν
are
solutions to the Bessel equation with parameter ν
z
2
u
+ zu
+(z
2
− ν
2
)u =0.
J
ν
(z)=(z/2)
ν
∞
k=0
(−z
2
/4)
k
k! Γ (ν + k +1)
N
ν
(z)=
J
ν
(z)cos(νπ) − J
−ν
(z)
sin(νπ)
.
The modified Bessel functions I
ν
and K
ν
are solutions to the Bessel equation
with parameter ν
z
2
u
(z)+zu
(z) − (z
2
+ ν
2
)u(z)=0
and are given by:
I
ν
(z)=
z
2
ν
∞
n=0
z
2n
2
2n
n! Γ (ν + n +1)
K
ν
(z)=
π(I
−ν
(z) − I
ν
(z))
2sinπν
.
Some recurrence relations:
d
dz
(z
ν
I
ν
(z)) = z
ν
I
ν−1
(z)
d
dz
(z
ν
K
ν
(z)) = −z
ν
K
ν−1
(z)
d
dz
(z
−ν
I
ν
(z)) = z
−ν
I
ν+1
(z)
d
dz
(z
−ν
K
ν
(z)) = −z
−ν
K
ν+1
(z)
I
ν−1
(z) − I
ν+1
(z)=
2ν
z
I
ν
(z) K
ν−1
(z) − K
ν+1
(z)=−
2ν
z
K
ν
(z)
(A.5.1)
A.5 Special Functions 663
The Bessel function K
ν
can be written in integral form
K
ν
(x)=
1
2
∞
0
y
ν−1
exp
−
1
2
x
y +
1
y
dy, x > 0 . (A.5.2)
Explicit formulae for some values of ν:
I
1/2
(z)=
2/πz sinh z,I
3/2
(z)=
2/πz (cosh z − z
−1
sinh z)
I
5/2
(z)=
2/πz ((1 + 3z
−2
)sinhz − 3z
−1
sinh z)
K
1/2
(z)=
2
πz
e
−z
,K
3/2
(z)=
2
πz
1+
1
z
e
−z
. (A.5.3)
A.5.3 Hermite Functions
The equation
u
− 2zu
+2νu =0
admits fundamental solutions of the form H
ν
(±z)where
H
ν
(z)=
1
2Γ (−ν)
∞
k=0
(−1)
k
k!
Γ
k − ν
2
(2z)
k
.
A.5.4 Parabolic Cylinder Functions
The Weber equation
u
+
ν +
1
2
−
z
2
4
u =0
has as a particular solution the parabolic cylinder function D
ν
(z)where
D
ν
(z) = exp
−
z
2
4
2
−ν/2
√
π H
ν
(z/
√
2)
and H
ν
is the Hermite function.
D
−ν
(z)=−
z
2
D
−ν
(z) − νD
−ν−1
(z) .
A.5.5 Airy Function
The Airy function (Ai)(x) is a solution of the Sturm-Liouville equation
u
(x)=xu(x),u(0) = 1 ,
and is given by
(Ai)(x)=
1
π
x
3
1/2
K
1/3
2
3
x
3/2
.
664 A List of Special Features, Probability Laws, and Functions
A.5.6 Kummer Functions
The Kummer functions M(α, γ; x)andU (α, γ; x) are solutions of
xu
+(γ − x)u
− αu =0
and are given by
M(α, γ; x)=
k
(α)
k
(γ)
k
x
k
k!
(A.5.4)
U(α, γ; x)=
π
sin πγ
M(α, γ; x)
Γ (1 + α − γ)Γ (γ)
+ x
1−γ
M(1 + α − γ, 2 − γ; x)
Γ (α)Γ(2 − γ)
where (α)
k
= α(α +1)···(α + k − 1) (See Lebedev [570] and Slater [803].)
The Kummer function M is also called the confluent hypergeometric
function, and is denoted
1
F
1
.When0<α<γ, one has
M(α, γ; x)=
Γ (γ)
Γ (γ − α)Γ (α)
1
0
e
xt
t
α−1
(1 − t)
γ−α−1
dt
U(α, γ; x)=
1
Γ (α)
∞
0
e
−xt
t
α−1
(1 + t)
γ−α−1
dt ,
The solutions of
x
2
u
+(αx +1)u
= λu
are Φ
λ↑
(x)andΦ
λ↓
(x) defined as
Φ
λ↑
(x)=
1
x
(ν+μ)/2
M
ν + μ
2
, 1+μ,
1
x
Φ
λ↓
(x)=
1
x
(ν+μ)/2
U
ν + μ
2
, 1+μ,
1
x
where M and U denote the Kummer functions and μ =
√
ν
2
+4λ,1+ν = α.
It follows that the solutions of
x
2
u
+(αx + β)u
= λu
are Φ
λ↑
(x/β)andΦ
λ↓
(x/β).
A.5.7 Whittaker Functions
The Whittaker functions W
k,m
and M
k,m
are solutions of the second order
differential equation
u
+
−
1
4
+
k
x
−
m
2
− 1/4
x
2
u =0.
Whittaker functions are related to the Kummer functions:
W
k,m
(x)=e
−x/2
x
m+1/2
U(m − k +1/2; 1 + 2m; x)
M
k,m
(x)=e
−x/2
x
m+1/2
M(m − k +1/2; 1 + 2m; x)
A.5 Special Functions 665
A.5.8 Some Laplace Transforms
∞
0
exp(−λx)f(x)dx = Ψ(λ) . (A.5.5)
Function Laplace transform
x
α−1
Γ (α)
λ
α
1
√
2πx
3/2
n∈Z
(v − u +2nv)e
−(v−u+2nv)
2
/2x
sinh(u
√
2λ)
λ sinh(v
√
2λ)
π
2
b
2
n≥1
(−1)
n+1
n
2
e
−n
2
π
2
x/(2b
2
)
b
√
2λ
sinh(b
√
2λ)
|a|
√
2πx
3
exp(−
a
2
2x
)
e
−|a|
√
2λ
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