Wavelet Methods - Computational Methods for Quantitative Finance - page 165

{

ψ ,k :

≤ ≤ ∈∇ }

of V L , we need to compute the stiff-
ness matrix entries A ( ,k ),(,k) = a(ψ ,k ,ψ ,k ) . Since A is, in general, densely
populated, we use wavelet compression to reduce the number of non-zero entries to
O

Using the basis 0 L, k

(N L ) . We need the following smoothness assumption on the Lévy density k :

∂ n k(z) C 0 C n n | − α − 1 − n ,z | |≤ !| z = 0 , ∀ n ∈ N 0 , (12.7)

for C 0 ,C > 0. These kind of estimates are called Caldéron-Zygmund estimates.

12.2.2 Matrix Compression

The compression scheme is based on the fact that the matrix entries A ( ,k ),(,k) can
be estimated a priori and therefore neglected if these are smaller than some cut-off
parameter. There are two reasons for an entry to be omitted. Either the distance of
the supports supp ψ ,k and supp ψ ,k or the distance of the singular supports (the
singular support of a wavelet is that subset of G where the wavelet is not smooth) is
large enough. The distance of support is denoted by

δ := dist { supp ψ ,k , supp ψ ,k } ,

and the distance of singular support

dist

if ≤ ,

{ singsupp ψ ,k , supp ψ ,k }

δ sing :=

dist { supp ψ ,k , singsupp ψ ,k } else.

Theorem 12.2.2 Let X be a Lévy process with Lévy density k satisfying ( 12.7 ).
Define the compression scheme by

⎧
⎨
⎩

0 if δ> B , ,

A ( ,k ),(,k) =

if δ<c 2 − min { , } and δ sing > B , ,

0

A ( ,k ),(,k) , else,

with cut-of parameter

a max 2 − min { , } , 2

,a> 1 ,

2 L(p − α/ 2 ) − ( + )(p + p)
2

B , =

p + α

a max 2 − max { , } , 2

,

2 L(p − α/ 2 ) − ( + )p − max { , } p

B , =

a> 1 .

p + α

The number of non-zero entries for the compressed matrix A is

O (N L ) .

A proof can be found in [51, Theorem 11.1]. We call a compression scheme first
compression if δ>

B , , i.e. if the distance of the wavelets is large enough, and
second compression if δ sing > B , (and δ<c 2 − min { , } ), i.e. if the distance of the
singular support is large enough (compare with Fig. 12.2 ).
We give an example for the matrix compression.