Multidimensional Feller Processes - Computational Methods for Quantitative Finance - page 261

where X is a Feller process with characteristic triple (b(x), Q(x), k(x, z) d z) under
a risk neutral measure such that e X Q is a martingale with respect to the canonical

filtration of X . In the following we set r 0 for notational convenience. We consider

Feller processes X that are admissible time-homogeneous market models. In the
following we consider a special family of Feller processes to confirm the theoretical
results of the previous chapters.

=

Example 16.6.1 We consider a CGMY-type Feller process with jump kernel

C e − β + z z − 1 − Y(x) ,

z 0 ,

ke − x 2

k(x,z) = Y(x) = + 0 . 5 .

e − β − | z | | | − 1 − Y(x) ,z< 0 ,

z

This process has no Gaussian component and the drift γ(x) is chosen according to
( 16.23 ).

We also consider the following family of processes that do not satisfy the con-
ditions of the theory developed above, since the variable order is assumed to be
Lipschitz continuous only.

Example 16.6.2 We consider again a CGMY-type Feller process with jump kernel

C e − β + z z − 1 − Y(x) ,

z 0 ,

k(x,z) =

e − β − | z | |

| − 1 − Y(x) ,z< 0 , z

⎧
⎨

0 . 4 x, 0 . 25 >x> 0 ,

0 . 8 x − 0 . 1 , 0 . 5 >x ≥ 0 . 25 ,

Y(x) = 0 . 5 + k − 0 . 4 x + 0 . 5 , 0 . 75 >x ≥ 0 . 5 ,

⎩

− 0 . 8 x + 0 . 8 , 1 >x ≥ 0 . 75 ,

0 , else .
This process has no Gaussian component and the drift b(x) is chosen according
to ( 16.23 ).

In Fig. 16.1 the stiffness matrix for the process in Example 16.6.1 is depicted.
Note that the uncompressed stiffness matrix is densely populated, but structurally
very similar to the matrix in the Black-Scholes model, compare with Fig. 16.2 .
The assembly of the stiffness matrix is carried out using numerical quadratures.
Standard quadratures, such as the Gauss quadrature, yield unsatisfactory results due
to the weak singularity of the integrand. We therefore employ tensorized composite
Gauss quadratures for the computation of the matrix entries, see [138, Sect. 7.2],
[163, Chap. 5] and [149]. Using such an approach we obtain quadrature rules that
converge exponentially with respect to the number of quadrature points even for
weakly singular kernels. The condition numbers of the preconditioned stiffness ma-
trices have to be uniformly bounded in the number of levels due to arguments sim-
ilar to Sect. 12.2.3, we refer to [135, Chap. 5] for details. A parameter study for