Multidimensional Lévy Models - Computational Methods for Quantitative Finance - page 217

Fig. 14.7 Convergence rate
of the wavelet discretization
in terms of the mesh width h
( top )andintermsofdegrees
of freedom ( bottom )

where N ε is a compound Poisson process with jump measure ν ε . The small jump
part X ε is independent of N ε Q ε =

d zz ν ε ( d z) .

and has the covariance matrix

R

Q ε is non-singular. Let Σ ε be a non-singular matrix such that Σ ε Σ ε =
Q ε . X ε can be approximated by a d -dimensional standard Brownian motion W
independent of N ε . The next theorem [39, Theorem 3.1] shows that the process
Σ − 1
ε

We assume

X ε converges in distribution to W as ε → 0.

Theorem 14.7.1 Let X be a d-dimensional Lévy process with characteristic triplet
( 0 ,ν,γ) . Assume that Q ε is non-singular for every ε ∈ ( 0 , 1 ] and that for every

δ> 0 there holds

Q − ε z,z >δ Q − 1

z, z ν ε ( d z) → 0 , as ε → 0 .

ε