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

cf. lecture (107.241)

Problems to be solved by students

Presentation of solutions

Wie für 107.241