Nach positiver Absolvierung der Lehrveranstaltung sind Studierende in der Lage...
After successful completion of the course, students are able to...
- define stochastic processes, Markov chains, filtrations, stopping times
- define transition function and homogeneity of Markov processes
- define transition matrices and use them in calculations
- define and check successor and communicating relations
- define and chack period and recurrence properties
- define continuous time Markov chains
- define infinitesimal parameters of a continuous time Markov chain
- define Kolmogorv's differential equations and discuss their validity
- define the embedded discrete-time Markov chain and calculate its transition probabilities
- define and calculate the infinitesimal operator
- understand Markov chain mixing via spectral gap
- define martingales, sub- and supermartingales
- discuss the influence of transformations on the martingale property
- cite and apply the optional stopping and optional selection theorems
- cite and apply Doob's maximum inequalities
- cite and apply the martingale convergence theorem
- define the Doob-Meyer decomposition and discuss its existence
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
