Multidimensional Feller Processes - Computational Methods for Quantitative Finance - page 255

Definition 16.4.1 We call a d -dimensional Feller process with characteristic triplet
(γ (x), Q(x), ν(x, d z)) a time-homogeneous admissible market model if it satisfies
the following properties.

d

(i) The vector function x → b(x) ∈ R is smooth and bounded.

d d

sym

×

→ ∈ R

(ii) The matrix function x Q(x) is smooth and bounded and, for all

d , the matrix Q(x) is positive semidefinite.
(iii) The jump measure ν(x, d z) is constructed from d independent, univariate
Feller-Lévy measures with a 1-homogeneous copula function F that fulfills
the following estimate: there is a constant C> 0 such that for all u

∈ R

x

) d ∈ ( R\{ 0 }

0

and all n ∈ N it holds

d

∂ n F(u) ≤ C | n |+ 1

1 | u i | − n i .

| n |! min {| u 1 | ,..., | u d |}

i =

(iv) For the marginal densities ν i (x i , d z i ) = k i (x, z) d z the mapping x i → ν i (x i ,B)

) .
(v) There exist univariate Lévy kernels k i (z) , i

is smooth for all B ∈ B ( R

= 1 ,...,d , with semi-heavy tails,

i.e. which satisfy

C e − β − | z | ,z< −

1 ,

≤

k i (z) (16.19)

e − β + z , z> 1 ,

for some constants C> 0, β − > 0 and β + > 1. These Lévy kernels satisfy the
following estimates

0 ≤ ν i (x i ,B) ≤ k i (z) d z ∀ x i ∈ R ,B ∈ B ( R ), i = 1 ,...,d.

B

(vi) Besides, we require the following estimates on the derivatives of k i (x, z)
∂ x k i (x, z) ≤ C n + 1 n ! | z | − Y i (x) − δn − 1 ,
∂ z k i (x, z) ≤ C n + 1 n ! | z | − Y i (x) − n − 1 ,

for any δ ∈ ( 0 , 1 ) , for all 0 = z, x ∈ R and Y< 2aswellas Y > 0, Y i (x) =

Y i + Y i (x) , Y i ∈ R +

and Y i (x) 1 ,...,d .

(vii) Finally, we require F 0 to be a 1-homogeneous Lévy copula and k i (x i ,z i ) to
be Y i (x i ) -stable densities with tail integrals U i (x i ,z i ) , i

∈ S ( R ) , i =

= 1 ,...,d such that

Ck i (x, z), k i (x, z) ≥ ∀ 0 < | z | < 1 , ∀ x ∈ R ,i = 1 ,...,d,

∂ n F 0 (U 0 (x, z)) ∂ 1 ··· ∂ n F(U(x,z)) ≥ C∂ 1 ··· ∀ 0 < | z | < 1 ,

for some constant C> 0.

=

Remark 16.4.2 Note that the smoothness assumptions on Y i (x i ) , i 1 ,...,d ,are

necessary in order to obtain symbols as given in Definition 16.2.2 . We confine the
discussion to such symbols in order to use the results of [137] that rely on pseu-
dodifferential calculus for symbols of variable order, cf. [86, 104]. The derivation
of similar results for symbols with lower regularity is open to our knowledge.