After successful completion of the course, students are able to

• define stochastic processes,
• define and identify certain types of processes
• define filtrations,
• define and identify stopping times
• define transition function and homogeneity of Markov processes
• cite the Capman-Kolmogorov equations
• define Markov chains
• 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
• define and analyze birth- and death processes
• define explosion and regularity and give criteria for regulariy
• define the minimal solution
• discuss path properties of general Markov processes
• define the transition operators of a Markov process and discuss their properties
• define the resolvent and discuss its properties
• define and calculate the infiniresimal operator
• cite the Hille-Yosida theorem
• 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
• define the Ito integral
• define and apply Ito's formula
• define stochastic differential equations
• cite the existence and uniqueness theorem for stochastic differential equations

general theory; 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, conservative chains, Kolmogorov differential equations, embedded discrete Markov chain, bitrh-and death processes, explosion, regularity, minimal solution; general theory of Markov processes: path properties, transition oprators, resolvent, infinitesimal operator, Hille-Yosida theorem; martingales: definition, semimartingales, transformations, optional stopping, optional selection, maximum inequality, martingale convergence theorem, Doob-Meyer decomposition; stochastic calculus: Ito integral, Ito's formula, stochastic differential equations: existebce and uniqueness theorem

chalk, blackboard, voice

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