Some investigations on Feller processes generated by pseudo-differential operators. - Bjorn Bottcher (2004)
Introducing an appropriate symbolic calculus for non-classical real-valued symbols, so-called negative definite symbols, W. Hoh succeeded to prove that such operators generate often Feller semigroups. In a first part of this thesis we extend this result to complex-valued symbols. Further, using ideas due to H. Kumano-go in case of classical pseudo-differential operators, we construct a parametrix for the fundamental solution of the associated evolution equation, and thus arrive at an approximation for the generated Feller semigroup. Finally, we use this theory to extend models in financial mathematics based on Levy processes. This is done by using the above mentioned results in situations where parameters in characteristic exponents of Levy processes are made state- space dependent. Especially Meixner-type processes are discussed in detail.