Lévy Models - Computational Methods for Quantitative Finance - page 143

10.8 Further Reading

For a comprehensive introduction to Lévy processes, we refer to Sato [143] and
Bertoin [18]. An overview for various Lévy models is given in Cont and Tankov [40]
and Schoutens [148]. A similar localization argument to Theorem 10.5.1 is given by
Cont and Voltchkova in [41] under stronger assumptions on the initial condition.
A finite difference method for the discretization of the PIDE similar to Sect. 10.6.1
was proposed by Biswas [20]. In Cont and Voltchkova [41, 42], small jumps were
approximated by an artificial diffusion and a finite difference discretization of the
PIDE with small jumps truncated was proposed in this case. Similar techniques for
d

2 are also shown in Briani et al. [28].
Due to the non-locality of the integro-differential operator in the infinitesimal
generator of a Lévy process, the corresponding stiffness matrix becomes fully pop-
ulated, no matter whether we use a finite difference or a finite element discretization.
In the latter case, however, one can “compress” the matrix to a sparsely populated
matrix, without loss of asymptotic convergence rate of discretization errors, pro-
vided the basis functions which span the finite element space are properly chosen.
One instance of such basis functions are wavelets, which we will introduce and
explain in Chap. 12.
Lévy copulas [100] are used for parametric constructions of Lévy processes in
d -dimensions. These models are explained in detail in Chap. 14.

= 1 and d =