The core topic of the lecture are Markov Processes in continuous time. We will begin with a preparation of the setting of continuous time stochastic processes on the space of càdlàg functions, in particular the theory of continuous time martingales. The general theory of Markov processes will have to main components. First the theory of strongly continous Markov semigroups and its connection to the notion of the generator, culminating in the Hille-Yoshida theorem and the construction of Feller-Dynkin processes. The second approach will be through the so-called martingale problem. This approach will bear fruits in the applications, in particular in the context of stochastic differential equations.
There are lecture notes available here.
