Multidimensional Feller Processes - Computational Methods for Quantitative Finance - page 249

∈ S ∗ (

d ) , where

S ∗ (

d ) denotes the space of tempered

≥ R R

non-integer s 0 and u

distributions,

2 ) s ˆ

u(ξ )

2

2
H s (

+ |

|

u

d ) :=

( 1 ξ d ξ. (16.8)

R

d

R

Similarly, we can define anisotropic Sobolev spaces H s (

d ) with norm

R · H s

given by

d

ξ j ) s j ˆ u(ξ )

2

2
H s (

u

d ) :=

( 1 + d ξ, (16.9)

R

d

R

j = 1

for any multi-index s 0. The consideration of certain symbol classes will be useful

for the definition of the variable order Sobolev spaces. We set

≥

2 ) 1 / 2

ξ := ( 1 +| ξ |

for notational convenience.

C ∞ (

d ) be a real-valued

Definition 16.2.1 Let 0 ≤ δ<ρ ≤ 1 and let m(x) ∈ R

d

function with bounded derivatives on R of arbitrary order. Then, the symbol

ψ(x,ξ) belongs to the class S m(x)
ρ,δ

of symbols of variable order m(x) if ψ(x,ξ) ∈

C ∞ (

d d ) and m(x) d ) a tempered function, and if,

R × R = + ∈ S R

s m(x) with m (

d
0

for every α, β ∈ N there is a constant c α,β such that

d

D x D ξ ψ(x,ξ)

m(x) − ρ | α |+ δ | β | .

∀ x,ξ ∈ R : | |≤ c α,β ξ (16.10)

Ψ m(x)
ρ,δ

The variable order pseudodifferential operators A (x, D) ∈ correspond to

S m(x)
ρ,δ

symbols ψ(x,ξ) ∈ by

1
2 π

e i x − y,ξ ψ(x,ξ)u(y) d y d ξ, C 0

d ).
(16.11)

A := ∈ R

(x, D)u(x) u (

d d

R R

We are now able to define an isotropic Sobolev space of variable order
H m(x) (

d ) , m(x)

0, using the variable order Riesz potential Λ m(x) R ≥ with sym-

m(x) . Clearly, ψ(x,ξ) is an element of S m(x)
1 ,δ

bol ψ(x,ξ) = ξ for δ ∈ ( 0 , 1 ) .The

norm on H m(x) (

d ) is given as

R

2
H m(x) (

Λ 2 m(x) u

2
L 2 ( R

2
L 2 ( R

u

d ) := d ) +

u

d ) .

R

1, we obtain the usual L 2 (

d ) -norm. For ψ(x,ξ)

Note that for ψ(x,ξ) = R = ( 1 +

s ) , we obtain the norm given in ( 16.8 ), which follows by applying Plancherel's
theorem. Now we turn to the definition of anisotropic variable order Sobolev spaces.
In analogy to Definition 16.2.1 , we start with the definition of an appropriate symbol
class.

| ξ |

d d

Definition 16.2.2 Let m (x) = s + m (x) , m (x) : R → R with each component of

d
+

m (x) being a tempered function and s ∈ R ,0 ≤ δ<ρ ≤ 1. We define the symbol