probability and stochastic processes

Lectures
  • (18-9-2018) General introduction, probability spaces, examples, continuity of the probability, first Borel Cantelli lemma.
  • (19-9-2018) Random variables, sigma algebras generated by families of events and random variables, distributions, simple random variables.
  • (25-9-2018) Abstract integration on measurable spaces(short hints), expected values, integrable random variables, change of variables formula for integration examples
  • (26-9-2018) Independence of sigma algebras, second Borel Cantelli lemma, example and exercises
  • (2-10-2018) Conditional expectations with respect to an event, conditional expectations with respect to a discrete random variable.
  • (3-10-2018)General definition of conditinal expectation with respect to a sigma algebra, existence (only statement), uniqueness, examples
  • (9-10-2018) Proofs of the properties of conditional expectations, conditional Jensen inequality, conditional expected value as a projection, filtrations, general definition of martingales, submartingales, supermartingales
  • (10-10-2018) Examples of martingales, Doob martingales, gambling strategies, martingale transform and preservation of the martingale property, stopping times.
  • (16-10-2018) Stopped processes and the process observed at the stopping time, stopped martingales are martingales, The martingale
  • (17-10-2018) The optional stopping Theorem, application to the exit from an interval of a random walk, the symmetric and the asymmetric case, Markov chains and conditional expectations
  • (23-10-2018) Remarkable example: mean time of hitting of a given pattern, Branching processes, Polya urn.
  • (24-10-2018) Maximal Doob inequality, example and exercises

    textbooks
  • Main textbook: Zdzislaw Brzezniak, Tomasz Zastawniak: Basic stochastic processes (a course through exercises)
  • Suggested book for insights: D. Williams: Probability with martingales,
  • Useful material can be found in the lecture notes by Peter Morters

    Past exams
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    News
  • Partial examination on Tuesday 6 of November at 14.30 room A.1.3