107.241 Theory of stochastic processes
This course is in all assigned curricula part of the STEOP.
This course is in at least 1 assigned curriculum part of the STEOP.

2020S, VO, 3.0h, 5.0EC
TUWEL

Properties

  • Semester hours: 3.0
  • Credits: 5.0
  • Type: VO Lecture

Learning outcomes

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

Subject of course

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

Teaching methods

chalk, blackboard, voice

 

Mode of examination

Oral

Lecturers

Institute

Course dates

DayTimeDateLocationDescription
Wed11:30 - 14:0011.03.2020Sem.R. DA grün 06A TSP Mittwoch

Examination modalities

exam

Course registration

Begin End Deregistration end
02.03.2020 00:00 02.04.2020 00:00

Curricula

Literature

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

Previous knowledge

Measure and probability theory

Miscellaneous

Language

if required in English