Wavelet Methods - Computational Methods for Quantitative Finance - page 160

Fig. 12.1 Single-scale space
V L and its decomposition into
multiscale wavelet spaces W
for L = 3

(ii) Iterative solution and judicious preconditioning of the linear systems in each
time step,
(iii) Wavelet finite element bases for the preconditioning and the matrix compres-
sion of the nonlocal operators arising with jump type price processes.

We start with explaining spline wavelets on an interval.

12.1 Spline Wavelets

As in Sect. 3.3, we discretize the domain G = (a, b) by equidistant mesh points

= ···

<x N L + 1 =

a x 0 <x 1 <x 2 < b,

2 L + 1

where we assume the number N L satisfies N L = − 1 with L ∈ N 0 and use the

V N L . Then, we have the nested spaces with 2 , 4 ,..., 2 L + 1

notation V L = subinter-

vals

V 0 ⊂ V 1 ⊂···⊂ V L ,

2 + 1

and dim V = − 1 =: N . In Sect. 3.3, we generated V by a single-scale

basis V = . Here, we change notation and we write

b ,j instead of b j to indicate the refinement level. Wavelets constitute a so-
called hierarchical or multi-scale basis. We start with

span { b ,j (x) : 1 ≤ j ≤ N }

{ ψ 0 , 1 } for the space V 0 .

Then, we add basis functions { ψ 1 , 1 ,ψ 1 , 2 } such that span { ψ 0 , 1 ,ψ 1 , 1 ,ψ 1 , 2 }=

{

ψ 2 , 1 ,ψ 2 , 2 ,ψ 2 , 3 ,ψ 2 , 4 }

V 1 . Similarly, we add again basis functions such that

span V 2 , and so on. Therefore, we in-

troduce for ∈ N 0 the complement spaces W =

{ ψ 0 , 1 ,ψ 1 , 1 ,ψ 1 , 2 ,ψ 2 , 1 ,ψ 2 , 1 ,ψ 2 , 3 ,ψ 2 , 4 }=

span { ψ ,k : k ∈∇ } where

1 ,..., 2

∇ := { } such that V = V − 1 ⊕ W , ≥ 1 and V 0 = W 0 . This decom-

position is illustrated in Fig. 12.1 .
We assume that the wavelets ψ ,k have compact support

C 2 − ,

| supp ψ ,k |≤

are normalized in L 2 (G) ,i.e. ψ ,k L 2 = 1, and Φ := { b ,j (x) : 1 ≤ j ≤ N } has