Mathematical and Statistical Methods for Actuarial Sciences and Finance

Tempered stable distributions and processes in finance: numerical analysis 35

Furthermore, by theorem 2.3 in [21], the Levy measure ́ νcan be also rewritten in
the form

ν(A)=

∫

Rd 0

∫∞

0

IA(tx)αt−α−^1 e−tdt R(dx), A∈B(Rd), (2)

whereRis a unique measure onRdsuch thatR({ 0 })= 0
∫

Rd

(‖x‖^2 ∧‖x‖α)R(dx)<∞,α∈( 0 , 2 ). (3)

Sometimes the only knowledge of the L ́evy measure cannot be enough to obtain
analytical properties of tempered stable distributions. Therefore, the definition of
Rosi ́nski measureRallows one to overcome this problem and to obtain explicit ana-
lytic formulas and more explicit calculations. For instance, the characteristic function
can be rewritten by directly using the measureRinstead ofν(see theorem 2.9 in [21]).
Of course, given a measureRit is always possible to find the corresponding tempering
functionq; the converse is true as well. As a consequence of this, the specification of
a measureRsatisfying conditions (3), or the specification of a completely monotone
functionq, uniquely defines a TSαdistribution.
Now, let us define two parametric examples. In the first example the measureRis
the sum of two Dirac measures multiplied for opportune constants, while the spectral
measureRof the second example has a nontrivial bounded support. If we set

q(r,± 1 )=e−λ±r,λ> 0 , (4)

and the measure
σ({− 1 })=c− and σ({ 1 })=c+, (5)

we get

ν(dr)=

c−

|r|^1 +α−

e−λ−rI{r< 0 }+

c+

|r|^1 +α+

e−λ+rI{r> 0 }. (6)

The measuresQandRare given by

Q=c−δ−λ−+c+δλ+ (7)

and
R=c−λα−δ− 1
λ−
+c+λα+δ 1
λ+

, (8)

whereδλis the Dirac measure atλ(see [21] for the definition of the measureQ).
Then the characteristic exponent has the form

ψ(u)=iub+(−α)c+((λ+−iu)α−λα++iαλα+−^1 u)

+(−α)c−((λ−+iu)α−λα−−iαλα−−^1 u),

(9)

where we are considering the L ́evy-Khinchin formula with truncation functionh(x)=
x. This distribution is usually referred to as the KoBoL or generalised tempered stable
(GTS) distribution. If we takeλ+=M,λ−=G,c+=c−=C,α=Yandm=b,
we obtain thatXis CGMY distributed with expected valuem. The definition of the
corresponding Levy process follows. ́