Borders of the probabilistic symbol, time-inhomogeneity, and generalized semimartingales

When investigating semimartingales, both the characteristics and the probabilistic symbol play important roles. They allow for the analysis of various significant properties and, in some cases, the characterization of the underlying process. For instance, the symbol, which is related to the right derivative of the characteristic function of the one-dimensional marginals, of a Lévy process, coincides with the characteristic exponent, and for Feller processes with the symbol of the operator. The most general class for which the symbol exists is Itô processes.
In this thesis, we show that within the class of Hunt semimartingales, Itô processes are precisely those for which the probabilistic symbol exists. Furthermore, we point out that the applicability of the symbol can be lost for processes that are not Hunt semimartingales, even if the symbol exists.
Investigating beyond time-homogeneity, we add a time component to the symbol to analyze non-homogeneous processes. We show the existence of such a time-dependent symbol for non-homogeneous Itô processes. Additionally, for this class of processes, we derive maximal inequalities, which we apply to extend the Blumenthal-Getoor indices to the non-homogeneous case. These allow for the derivation of various properties concerning the paths of the process.
Lastly, we generalize semimartingales by introducing a `point of no return' or `killing point', as known in the Markovian context, within this framework. To this end, the development of a new characteristic to describe this phenomenon is required. We present a theory concerning these generalized semimartingales by extending some of the most important classical results
with the help of the new characteristic, and integrate the probabilistic symbol into this context. Additionally, we present a natural way in which a killing can occur in the semimartingale framework.