Multidimensional Feller Processes - Computational Methods for Quantitative Finance - page 251

d . The spaces of order m (x)

R ≤

where u is the zero extension of u to all of 0,

d , are defined by duality. We have

∀ x ∈ R

( H − m (x) (G)) ∗ ,
where duality is understood with respect to the “pivot” space L 2 (G) ,i.e. L 2 (G) ∗ =
L 2 (G) .

H m (x) (G)

:=

Remark 16.2.3 In the Black-Scholes case, H 1 ( R

d ) is obtained as the domain of the
corresponding bilinear form while H 0 (G) is the domain in the localized case, see
Chap. 4. In the Lévy case, we obtain anisotropic Sobolev spaces as in ( 16.9 ) and the
spaces

H s (G) in the localized case for Q =

0, see Chap. 14.For Q> 0 the domains
are equal to those in the Black-Scholes case, cf. [134, Theorem 4.8].

Several examples of Feller processes are provided in this section. The methods
presented in this work are not applicable to all of them. But admissible market mod-
els are derived ensuring the well-posedness of the corresponding pricing equations
and the applicability of finite element methods. Subordination can be used to con-
struct Feller processes. However, the structure of the symbol is more involved than
in the Lévy setting. We also present a construction of Feller processes using Lévy
copulas.

16.3 Subordination

Many Lévy models in the context of option pricing are constructed via subordination
of a Brownian motion by a corresponding stochastic clock, e.g. an NIG process [8]
or a VG process [119]. We describe a similar construction for Feller processes and
point out similarities and differences to the Lévy case. Bernstein functions play a
crucial role in the representation of subordinators.

C ∞ ( 0 ,

Definition 16.3.1 A function f(x) ∈ ∞ ) is called a Bernstein function if

1 ) k ∂ k f(x)
∂x k

f ≥ 0 ,( − ≤ 0 , ∀ k ∈ N .

e − cx , c Example 16.3.2 The functions f 1 (x) = c , f 2 (x) = cx and f 3 (x) = 1 − ≥ 0

are Bernstein functions.

Bernstein functions admit the following representation.

Theorem 16.3.3 For any Bernstein function f(x) there exists a measure μ on
( 0 , ∞ ) , with

∞

s

∞

s μ( d s) < 1 +

0 +