Multidimensional Lévy Models - Computational Methods for Quantitative Finance - page 213

14.6.1 Wavelet Compression

To define the compression scheme, we need to introduce some notation. Consider
tensor product wavelets ψ , k =

ψ 1 ,k 1 ⊗···⊗ ψ d ,k d , ψ , k = ψ 1 ,k 1 ⊗···⊗ ψ d ,k d .

The distance of support in each coordinate direction is denoted by
δ x i := dist { supp ψ i ,k i , supp ψ i ,k i } ,

for i = 1 ,...,d , and the distance of singular support

dist

i , { singsupp ψ i ,k i , supp ψ i ,k i } if i ≤

δ sing :=

x i

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

Define

L(p

| |

− |

|

− |

| ∞

− α/ 2 ) − p if p(L ) ≥ α/ 2 (L ),

L , :=

| | ∞

− α/ 2 else

p

) ∞

L(p

− α/ 2 ) − if p(L − ≥ α/ 2 (L − ),

α/ 2 ∞

+ (14.24)

− else

and m i := i + i − { i , i }

c

2min . Furthermore, we denote the index sets I

, , I , ⊂

{ 1 ,...,d } by

i

δ x i > 2 − min { i , i } ,

c c

I

, =

∈{ 1 ,...,d }: I , ={ 1 ,...,d }\ I

, , (14.25)

and set
β i , = L , − p( i + i ) + α
j

m j − p
j

1
2

{ j , j }+ min m j ,

c
, \{

= i j ∈ I , ∈ I i }

{ i , i }+ α
j

m j − p
j ∈ I

1
2

β i , = L , − p max

{ j , j }+ min m j .

c
,

= i j ∈ I , \{ i }

(14.26)

The cut-off parameters are now defined by

a i max 2 − min { i , i } , 2 β i , /( 2 p + α) , i > 1 ,

i

B

, =

a i max 2 − max { i , i } , 2 β i , /(p + α) ,

B

i

, =

a i > 1 .

We define a multidimensional version of Theorem 12.2.2. A proof can be found in
[134, Theorem 4.6.3].

Theorem 14.6.1 Let X be a Lévy process with Lévy density k satisfying ( 14.23 ).
Define the compression scheme by

⎧
⎨

c
,

i

0 if ∃ i ∈ I : δ x i > B

, ,

δ sing

A ( , k ),( , k ) =

> B

i

∃ ∈ I , :

0 if i

, ,

x i

⎩

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