Implications of Markov stability theory for nonparametric statistics, Markov additive fluctuations and data-driven stochastic control

The central topic of this thesis is the influence of stability properties of continuous time Markov processes on their nonparametric statistical analysis. In particular sup-norm adaptive invariant density estimation under assumptions on the ergodic behavior of the process is investigated and consequently applied to jump diffusions with Lévy-driven jump part. Furthermore, the findings are used to demonstrate how statistical procedures for Markov processes can be implemented for the development of efficient data-driven strategies for stochastic optimal control problems associated to both continuous diffusion processes and Lévy processes. As one of the main theoretical tools in this regard, we give a detailed analysis of stability properties of overshoots associated to Markov additive processes. This allows incorporating fluctuation theory for Markov additive processes and Lévy processes into our general statistical framework, which is essential for the data-driven Lévy control strategy. Moreover, the overshoot analysis guides us naturally into extending some well-known fluctuation results for Lévy processes to the more general case of Markov additive processes and—making use of the one-to-one correspondence between Markov additive processes and real self-similar Markov processes through the Lamperti–Kiu transform—gives us the right tool to analyze the mixing behavior of self-similar Markov processes sampled at first hitting times