Multidimensional Lévy Models - Computational Methods for Quantitative Finance - page 208

Fig. 14.4 Anisotropic
α -stable Lévy copula
density ( 14.14 )in d = 2for
α = ( 0 . 5 , 1 . 2 ) and
corresponding contour plot

(ii) Furthermore, we assume there exist an α -stable process X 0 with Lévy density

k 0

and C > 0 such that

+

C + k 0 (z),

| |

k(z) ≤ 0 < z < 1 . (14.16)

(iii) If Q is not positive definite, we assume additionally that there is a C − > 0 such

that

k sym (z) ≥ C − k 0 , sym (z), 0 < | z | < 1 . (14.17)

1, the assumptions ( 14.15 )-( 14.17 ) coincide with the assumptions
(10.11)-(10.13). In the case that the marginal processes X i , i =

For d =

1 ,...,d are inde-
pendent, it is equivalent that the corresponding marginal one-dimensional densities
k i , i = 1 ,...,d satisfy (10.11)-(10.13).