Multidimensional Lévy Models - Computational Methods for Quantitative Finance - page 219

Fig. 14.8 Regularized
anisotropic α -stable Lévy
copula density for
α = ( 0 . 5 , 1 . 2 ) , ε = 0 . 01 and
corresponding contour plot

Q ε ,ν ε ,γ 2 ) . The processes W and N are independent. Y 2
has the same covariance matrix as X and drift γ 2 ,j =

with characteristic triplet (

γ 1 ,j − Q ε,jj / 2, j = 1 ,...,d .

we obtain a diffusion process Y t For ε →∞ = Σ W t + γ ∞ t with the covariance

ΣΣ matrix 1 ,...,d .

There are two sources of error. We have a discretization error using a mesh width
h> 0 and a modeling error using ε> 0. To assess the impact of ε> 0, we use
the wavelet discretization for ε

Q = and drift γ ∞ ,j =− Q jj / 2, j =

=

0 and ε> 0. Here, h is chosen so small that the
discretization error is negligible in comparison to the truncation error due to cut-off
of jumps of size smaller than ε .