Wavelet Methods - Computational Methods for Quantitative Finance - page 163

equivalences hold in the whole range of Sobolev spaces, i.e. from L 2 (G) to H 0 (G) .
For u L 2 (G) , there holds ∈

∞ ∞

k ∈∇ |

2 2 2 .

u

L 2 (G) ∼

0

u

L 2 (G) ∼

u ,k |

= =

0

L 2 (G) The mapping u → u 0 +···+ u defines a continuous projector P : → V .

H s (G) , we have the direct (or Jackson type) estimate,

For general Sobolev spaces

C 2 − ls

u − P u L 2 (G) ≤ u H s (G) , 0 ≤ s ≤ p.

For u ∈ V we also have the inverse (or Bernstein-type) estimates,

u H s (G) ≤ C 2 ls

u L 2 (G) ,s<p − 1 / 2 .

Using the inverse estimate and the series representation ( 12.1 ), we have

∞ ∞

2 2

2 2 ls

2 ,

u

H s (G) ∼

0 u

H s (G) ≤ C

| u ,k | 0 ≤ s<p − 1 / 2 .

= = 0 k ∈∇

∈ H p (G) In the other direction, we have for u

u L 2 (G) = P u − P − 1 u L 2 (G) ≤ P u − u L 2 (G) + P − 1 u − u L 2 (G)
≤ C( 2 − p 2 p 2 − p ) + u H p (G) ,

and therefore,

L

2 2 lp

2

C L

2

| u ,k | ≤ u

H p (G) .

= 0 k ∈∇

Unfortunately, we do not quite obtain the required bound. This estimate is sharp.
For 0 ≤ s<p , one can modify this argument by using the modulus of continuity

and obtain

∞ ∞

2 2

2 2 ls

2 ,

u

H s (G) ∼

0

u

H s (G) ∼

| u ,k | 0 ≤ s<p − 1 / 2 . (12.3)

= = 0 k ∈∇

12.2 Wavelet Discretization

Let X be a Lévy process with characteristic triplet ( 0 ,ν, 0 ) satisfying Assump-
tion 10.2.3. As shown in Chap. 10, the option price can be obtained by the solution
of the PIDE

∂ t u − A u = 0 n J × G, u( 0 ,x) = u 0 in G = ( − R,R), (12.4)