Multidimensional Feller Processes - Computational Methods for Quantitative Finance - page 245

Chapter 16
Multidimensional Feller Processes

In this chapter, we extend the setting of Chap. 14 to a more general class of pro-
cesses. We consider a large class of Markov processes in the following. Under cer-
tain assumptions we can apply the theory of pseudodifferential operators in order to
analyse the arising pricing equations. The dependence structure of the purely dis-
continuous part of the market model X is described using Lévy copulas. Wavelets
are used for the discretization and preconditioning of the arising PIDEs, which are
of variable order with the order depending on the jump state.

16.1 Pseudodifferential Operators

We introduce a class of stochastic processes which are characterized via the symbol
of their generator. Semimartingales are a well-investigated class of stochastic pro-
cesses that is sufficiently rich to include most of the stochastic processes commonly
employed in financial modelling while still being closed under various operations
such as conditional expectations and stopping. Semimartingales can be well under-
stood via their (generally stochastic) semimartingale characteristic, we refer to the
standard reference [97] for details. Here, we restrict ourselves to a class of processes
with deterministic, but generally state-space dependent characteristic triplets includ-
ing Lévy processes, affine processes and many local volatility models. We consider
an

d -valued Markov process X and the corresponding family of linear operators
(T s,t ) for 0

R

≤ s ≤ t< ∞ given by

(T s,t (f ))(x) = E[ f(X(t)) | X(s) = x ] ,

d ) , x d . Here, B b ( d ) denotes the space of bounded Borel

for each f ∈ B b ( R ∈ R R

d . In the following, we call a Markov process X normal
if its associated semigroup T satisfies

measurable functions on R

d )) d ).

T s,t (B b ( R ⊂ B b ( R (16.1)

We recall the following properties: