Some general theory of stochastic processes; types of stochastic processes, path properties, filtrations and stopping times,
Markov Processes: transition function, homogeneity, Chapman-Kolmogorov equations, Markov chains: transition matrices, successors, communicating states, period, recurrence properties, absorption, Markov chains in continuous time: infinitesimal parameters, Kolmogorov differential equations, embedded discrete Markov chain.
Reversible Markov chains, spectral analysis, spectral gap and relaxation time. If time allows: Markov chain mixing, path coupling method.
Martingales: definition, super- and sub-martingales, transformations, optional stopping, optional selection, maximal inequalities, martingale convergence theorem, Doob-Meyer decomposition, backward martingales with applications: Law of large numbers, de Finetti's theorem, Hewitt-Savage 0-1 law
