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The theoretical study of nonlinear expectations is the focus of attention for applications in a variety of different fields — often with the objective to model systems under incomplete information. Especially in mathematical finance, advances in the theory of sublinear expectations (also referred to as coherent risk measures) lay the theoretical foundation for modern approaches to evaluations under the presence of Knightian uncertainty. In this book, we introduce and study a large class of jump-type processes for sublinear expectations, which can be interpreted as Lévy-type processes under uncertainty in their characteristics. Moreover, we establish an existence and uniqueness theory for related nonlinear, nonlocal Hamilton-Jacobi-Bellman equations with non-dominated jump terms.

In this thesis, we present a new existence result for Lévy-type processes. Lévy-type processes behave locally like a Lévy process, but the Lévy triplet may depend on the current position of the process. They can be characterized by their so-called symbol; this is the analogue of the characteristic exponent in the Lévy case. Using a parametrix construction, we prove the existence of Lévy-type processes with a given symbol under weak regularity assumptions on the regularity of the symbol. Applications range from existence results for stable-like processes and mixed processes to uniqueness results for Lévy-driven stochastic differential equations. Moreover, we discuss sufficient conditions for the existence of moments of Lévy-type processes and derive estimates for fractional moments.

The concept of pseudo-differential operators allows one to study stochastic processes through their symbol. This approach has generated many new insights in recent years. However, most results are based on the assumption of bounded coefficients. In this thesis, we study Levy-type processes with unbounded coefficients and, especially, affine processes. In particular, we establish a connection between pseudo-differential operators and affine processes which are well-known from mathematical finance. Affine processes are an interesting example in this field since they have linearly growing and hence unbounded coefficients. New techniques and tools are developed to handle the affine case and then expanded to general Levy-type processes. In this way, the convergence of a simulation scheme based on a Markov chain approximation, results on path properties, and necessary conditions for the symmetry of operators were proven.

We characterize the structure of the domain of a pure jump-type Dirichlet form which is given by a Beurling–Deny formula. In particular, we obtain su cient conditions in terms of the jumping kernel guaranteeing that the test functions are a core for the Dirichlet form and that the form is a Silverstein extension. As an application we show that for recurrent Dirichlet forms the extended Dirichlet space can be interpreted in a natural way as a homogeneous Dirichlet space. For reflected Dirichlet spaces this leads to a simple purely analytic proof that the active reflected Dirichlet space (in the sense of Chen, Fukushima and Kuwae) coincides with the extended active reflected Dirichlet space., Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG-geförderten) Allianz- bzw. Nationallizenz frei zugänglich.

We prove that every (nice) Feller process is an It^o process in the sense of Cinlar, Jacod, Protter and Sharpe (1980). Next we generalize the notion of the symbol and define it for this larger class of processes. As examples the solutions of stochastic differential equations are considered. The symbol is then used to derive a quick approach to the semimartingale characteristics as well as the generator of the process under consideration. Finally we give some examples of how our methods work for processes used in mathematical finance., Wir haben gezeigt, dass jeder (nette) Feller Prozess ein It^o Prozess im Sinne von Cinlar, Jacod, Protter und Sharpe (1980) ist. Es stellt sich heraus, dass man den Begriff des Symbols, der für Feller Prozesse bekannt ist, auf diese größere Klasse verallgemeinern kann. Dieses Symbol haben wir für die Lösungen verschiedener stochastischer Differentialgleichungen berechnet. Außerdem haben wir gezeigt, dass das Symbol einen schnellen Zugang zur Berechnung der Semimartingal-Charakteristiken und des Erzeugers eines It^o Prozesses liefert. Zuletzt wurden die Ergebnisse auf Prozesse angewendet, die in der Finanzmathematik gebräuchlich sind. - (Die Dissertation ist veröffentlicht im Shaker Verlag GmbH, Postfach 101818, 52018 Aachen, Deutschland, http://www.shaker.de, ISBN: 978-3-8322-8244-8)