Stochastic Volatility Models with Jumps - Computational Methods for Quantitative Finance - page 237

∈ C

with λ . Again, we want to construct a diagonal preconditioner for the matrix B .

Since the underlying energy space

V (i.e. the space on which the bilinear form

a( is continuous and satisfies a Gårding inequality) for most of

the SV models is a weighted Sobolev space, we can no longer rely on the norm
equivalences (12.3), (13.8) to construct the preconditioners, but have to use norm
equivalences for weighted Sobolev spaces [19]. To this end, we recall some facts on
these.
The wavelets ψ ,k and the weighting function w defined on the interval ( 0 , 1 ) are
assumed to satisfy the following assumptions.

· , · ) : V × V → R

Assumption 15.4.1
(i) The wavelets have one vanishing moment: 0 ψ ,k (x) d x

0.
(ii) The wavelets ψ ,k and their duals ψ ,k belong to W 1 , ∞ ( 0 , 1 ) . Furthermore,
they satisfy: There exist constants C> 0, β,β

=

+ β>

∈ N 0 , γ + β> − 1 / 2, − γ

0 , 2 − − there holds

| (ψ ,k ) (j) (x) |≤ C 2 (j + 1 / 2 ) ( 2 x) β − j ,k ∈ I ,
| (ψ ,k ) (j) (x) |≤ C 2 (j + 1 / 2 ) ( 2 x) β − j ,j ∈ I ,
where the index sets

1 / 2, such that for j ∈{ 0 , 1 } and for x ∈[ ]

and I

2 − 1 , 0

I are given by I := { i ∈ N : β − 1 ≤ i ≤ ∈

and I := {

2 − 1 , 0 supp ψ ,i } , respectively.

(iii) The nonnegative weighting function w( · ) belongs to W 1 , ∞ (ε, 1 ) for every
ε> 0 and satisfies: there exists a constant C w > 0 such that for j

supp ψ ,i } i ∈ N : β − 1 ≤ i ≤ ∈

∈{ 0 , 1 }

w (j) (x)
x γ − j

C − 1
w

≤ ≤ C w , (15.24)

with γ ∈ R as in (ii).

The following weighted norm equivalences are proved in [19, Theorem 3.3, The-
orem 5.1].

Proposition 15.4.2 Let Assumption 15.4.1 hold and assume that (12.3) holds for
s = ∞ = 0 k ∈∇ u ,k ψ ,k and j

= 0. Then , for any u ∈{ 0 , 1 } the following norm

equivalence holds :

1
0 | u (j) (x) |

∞

u (j)

2 2 w 2 (x) d x

2 2 j w 2 ( 2 − k) | u ,k |

2 . (15.25)

L w ( 0 , 1 ) :=

= 0 k ∈∇

We need a tensorized version of Proposition 15.4.2 .

:= j = 1 w j (x j ) . Assume that w j : R →

≥

Corollary 15.4.3 Let d 1 and let w(x)

R + , j = 1 ,...,d , satisfies Assumption 15.4.1 (iii). Assume further that ψ , k (x) :=

ψ 1 ,k 1 ⊗···⊗ ψ d ,k d with ψ i ,k i satisfying Assumption 15.4.1 (i)-(ii). Then , for any

0

multi-index α ∈ N with | α | ∞ ≤ 1 and any

∞

=

u u , k ψ , k

i = 0 k i ∈∇ i