V3F1/F4F1 Stochastic Processes 

SS 2021

Tuesday 14.15-16.00 and Friday 10.15-12.00. Online via Zoom.

M. Gubinelli and L. La Rocca.

eCampus page


  (see eCampus)


  • 26/7-1/8 Oral exam, online.
  • Mid september. Oral exam, probably online.
  • Example questions for the oral exam (PDF)


The course is an introduction to various classes of stochastic processes, namely families of random variables indexed by a discrete or continuous parameter (time or space). Topic that will be covered include

  • General definition of conditional expectation
  • Martingales in discrete time and their convergence
  • Markov chains, their long time asymptotic and convergence to equilibrium    
  • Brownian motion: construction and sample path properties

The course is a follow-up of "Einführung in die Wahrscheinlichkeitstheorie” and a prerequisite to “Introduction to Stochastic Analysis" 


a good knowledge of measure theoretic probability as covered for example in:

  1. M. Gubinelli: Einführung in die Wahrscheinlichkeitstheorie, WiSe 2020/21 (link)
  2. R. Durrett: Probability: Theory & Examples, Chapters 1 and 2
  3. D. Williams: Probability with martingales, Part A and C

Lecture Notes

We will follow mainly the lectures notes of my previous course in SS2019: 

  • Note 1: Review of Measure spaces, Integration theory (v1.1, 10/4/19)
  • Note 2: Conditional expectations (v1.0, 15/4/19)
  • Note 3: Martingales (v1.1, 2/5/19)
  • Note 4: Asymptotic behavior of martingales (v1.1, 3/5/19)
  • Note 5: Closed martingales (v1.0, 13/5/19)
  • Note 6: Martingale CLT and some applications of martingales (Kolmogorov's LLN, Kakutani's theorem, Radon-Nikodym theorem) (v1.1, 4/6/219)
  • Note 7: Optimal stopping problems (v1.0, 3/6/19)
  • Note 8: Markov chains (v1.1, 18/6/19)
  • Note 9: Discrete chains, Doob's h-transform (v1.1, 21/7/16, updated version)

Further References

  • D. Williams: Probability with martingales, Part B.
  • R. Durrett: Probability: Theory & Examples.

Problem sheets

   (See on the eCampus page for the course)

Course Journal

  • Lecture 13/4 : Overview of the course. (script)
  • Lecture 16/4 : Review of measures spaces, Random variables, Dynkin's, Monotone class and Carathéodory extension theorems. (script)
  • Lecture 20/4: Integration. Lp spaces, completeness. Product measures and integrals, Fubini-Tonelli.  Uniform integrability (statements). (script)
  • Lecture 23/4: Uniform integrability (proofs). Conditional expectations: motivation.  (script)
  • Lecture 27/4: Definition and uniqueness of conditional expectations, examples.  (script)
  • Lecture 30/4:  Existence of conditional expectation via orthogonal projection in L^2. (script)
  • Lecture 4/5: Review of basic properties of conditional expectations, relation with independence, some examples. Proof of some properties of conditional expecations under independence assumptions.  (script)
  • Lecture 7/5: Regular conditional probabilities. Filtrations, adapted and previsible processes, stopping times. (script)
  • Lecture 11/5:  Sigma-algebra of a stopping time, Wald's identity for sums of independent random variables. (script)
  • Lecture 14/5: Martingales, some properties, Doob's decomposition. Quadratic variation. (script ,v.2)
  • Lecture 18/5: Martingale transform, Doob's optional sampling theorem, martingale property at stopping times. (script)
  • Lecture 21/5: Asymptotics of martingales, martingale convergence theorem. (script)
  • Lecture 1/6: Closed martingales. Square-integrable martingales. Doob's maximal and Lp inequalities. (script)
  • Lecture 4/6: Martingales closed in Lp and UI martingales. Optional stopping for closed martingales. (script)
  • Lecture 8/6: The tail sigma algebra and Kolmogorov's 0/1 law. Backwards martingales. Kolmogorov's law of large numbers. (script)
  • Lecture 11/6: Kakutani's theorem. (script)
  • Lecture 15/6: Radon-Nikodym theorem. (script)
  • Lecture 18/6: Optimal stopping problems. Value process, Snell's envelope and existence and characterisation of optimal stopping times. (script
  • Lecture 22/6: Markovian optimal stopping problems. Solution of Moser's problem. Random recurrences and the Markov property.  (script
  • Lecture 25/6: Markov processes. Transition kernel, some examples, general markov property. (script)
  • Lecture 29/6: Law of a Markov chain. Canonical space. Strong Markov property. (script)
  • Lecture 2/7: Martingale problems, relation with potential analysis, maximum principe. (script)
  • Lecture 6/7: Discrete chains, transience and recurrence.  (script)
  • Lecture 9/7: Computations of some probabilistic quantities via linear equations involving the generator. Doob's h-transform. (script)
  • Lecture 13/7: An application of the h-transform to conditioning of Markov chains. Invariant measures, existence and uniqueness.  (script)
  • Lecture 16/7: Conclusion of the discussion of invariant measures. Question session. (script)





Florian Schweiger erhielt den Hausdorff-Gedächtnispreis 2021 der Fachgruppe Mathematik für die beste Dissertation. Er fertigte die Dissertation unter der Betreuung von Prof. Stefan Müller an. Unter anderen wurde Vanessa Ryborz mit einem Preis der Bonner Mathematischen Gesellschaft für ihre von Prof. Sergio Conti betreute Bachelorarbeit ausgezeichnet. (18.01.2022)

Prof. Dr. Sergio Albeverio has been elected into the Academia Europaea and the Accademia Nazionale dei Lincei (more; 02.12.2021).

Der SFB 1060 Die Mathematik der emergenten Effekte hat eine dritte Förderperiode erhalten. (26.11.20)

Prof. Dr. Andreas Eberle erhält den diesjährigen Lehrpreis der Universität Bonn. (22.07.2020)

Herr Dr. Richard Höfer erhielt den Hausdorff-Gedächtnispreis 2019 der Fachgruppe Mathematik für die beste Disseration. Betreut wurde die Arbeit von Prof. J. Velázquez (29.01.2020).


Managing Director: Prof. Dr. Massimiliano Gubinelli
Chief Administrator: Dr. B. Doerffel
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Mailing address

Institute for Applied Mathematics
University of Bonn
Endenicher Allee 60
D-53115 Bonn / Germany