After successful completion of the course, students are able to

• define stochastic processes,
• define and identify Markov chains
• define filtrations,
• define and identify stopping times
• define transition function and homogeneity of Markov processes
• cite the Capman-Kolmogorov equation
• 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 conservative Markov chains
• define Kolmogorv's differential equations abd discuss their validity
• define the embedded discrete-time Markov chain and calculate its transition probabilities
• discuss path properties of general Markov processes
• define the transition operators of a Markov process and discuss their properties
• define and calculate the infiniresimal 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, semimartingales, transformations, optional stopping, optional selection, maximum inequality, martingale convergence theorem, Doob-Meyer decomposition

Via zoom

In case a majority of the students wishes to change schedule, I can see if I can find another solution.

oral exam

Bauer, H.: Wahrscheinlichkeitstheorie Neveu, J.: Martingales à temps discret

Karatzas, I.; Shreve, St.E.: Brownian motion and stochastic calculus

Rogers, L.C.G.; Williams, D.: Diffusions, Markov processes and martingales

Measure and probability theory