Abstracts

We consider a certain type of multidimensional normal Markov processes, which are Feller processes. This class of processes includes as special cases Lévy processes, many local volatility and local Lévy models. Due to their nonstationarity the considered processes can exhibit qualitative behaviour that is substantially different from that of Lévy processes such as state-space dependent jump activity. The nonstationarity also has substantial repercussions on their computational and analytical treatment: whereas for Lévy and the closely related affine models, Fast Fourier Transformation (FFT) algorithms form the basis for fast and powerful option pricing algorithms, the nonstationarity implies that FFT based numerical methods are, in general, not applicable in the numerical solution of their Kolmogorov equations (with the notable exception of, for example, affine processes).
From an analytical point of view, Feller processes are rather well understood. This is due to the fact that generators of Feller processes are pseudodifferential operators with symbols that admit a Lévy-Khintchine representation. Contrary to Lévy processes or diffusions with local volatility, domains of the infinitesimal generators for semigroups induced by Feller processes are, generally, variable order Sobolev spaces. Accordingly, the use of standard discretization schemes (based on Finite Differences or Finite Elements) for numerical solution of the Kolmogorov equations associated to such models is not straightforward; the same applies to the numerical analysis of these discretization schemes, i.e. the mathematical analysis of stability, consistency and convergence of these schemes.
One central theme of this talk is therefore to describe recent progress in the design and the numerical analysis of discretization schemes which allow a unified numerical treatment of the Kolmogorov equations for a large class of normal Markov processes. These schemes are based on variational, multiresolution schemes which use spline-wavelet bases of the domains of the processes' infinitesimal generators.