V3F1 Stochastic Processes
Tuesday 8.30-10.00 and Friday 8.30-10.00, Kleine Hörsaal, Wegelerstr. 10.
Exercise sheets will be distributed on Friday during the lecture and redactions collected on the next Thursday before 4pm unless specified otherwise. They will be the subject of the tutorials taking place the week after the collection of the redactions. Is possile to work in groups at most of two students. Tutorials will start the week of April 15th.
- Group 1: Mon 16-18 R0.006 - Manuel Esser <email@example.com> (in German)
- Group 2: Tue 12-14 R0.007 - Fran Medjurecan <firstname.lastname@example.org>
- Group 3: Thu 8-10 R0.006 - Daniel Klingmann <email@example.com>
- Group 4: Fri 14-16 R0.006 - Jona Lelmi <firstname.lastname@example.org>
Exercise sheets by: Immanuel Zachhuber and Nikolay Barashkov
In order to be allowed to partecipate to the exam session each student should hand in at least 80% of the exercise sheets and score at least 50% of the total of available points.
- Samstag 27.7.2019. From 9am to 11am in the Großer Hörsaal, Wegelerstraße 10.
- Donnerstag 19.9.2019. From 9am to 11am in the Kleiner Hörsaal, Wegelerstraße 10.
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:
- R. Durrett: Probability: Theory & Examples, Chapters 1 and 2
- D. Williams: Probability with martingales, Part A and C
- A. Eberle: Einführung in die Wahrscheinlichkeitstheorie, Lecture notes WT 2017/18, see wt.iam.uni-bonn.de/eberle/skripten/
We will follow mainly the lectures notes of Prof. Bovier SS2017 course (pdf). Detailed notes of the material presented in the lectures will be posted here:
- 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 (v1.0, 19/6/19)
- Note 10: Construction of stochastic processes. Daniell-Kolmogorov theorem (v1.0, 11/7/19)
- Note 11: Brownian motion and some of its properties (by F. de Vecchi, v1.0, 11/7/19)
- D. Williams: Probability with martingales, Part B.
- R. Durrett: Probability: Theory & Examples.
- Sheet 1 (due before 4pm on 11/4, revised version 9/4)
- Sheet 2 (due before 4pm on 18/4)
- Sheet 3 (due before 4pm on 25/4)
- Sheet 4 (due before 4pm on 2/5)
- Sheet 5 (due before 4pm on 9/5)
- Sheet 6 (due before 4pm on 16/5)
- Sheet 7 (due before 4pm on 23/5)
- Sheet 8 (due before noon on 31/5)
- Sheet 9 (due before 4pm on 6/6) (revised 4/6/19 to correct some typos)
- Sheet 10 (due before noon on 21/6)
- Sheet 11 (due before 4pm on 27/6)
- Sheet 12 (due before 4pm on 4/7)
- Lecture 2/4 : Overview of the course. Review of measures spaces, Carathéodory extension theorem.
- Lecture 5/4 : Random variables and integration. Uniform integrability.
- Lecture 9/4: Lp spaces, completeness, product measures and integrals, Fubini-Tonelli. Conditional expectations: motivation.
- Lecture 12/4: Definition and existence of conditional expectations.
- Lecture 16/4: Basic properties of conditional expectations, relation with independence, some examples.
- Lecture 19/4: (not taking place, Karfreitag)
- Lecture 23/4: Regular conditional probabilities, proof of some properties of conditional expecations under independence assumptions. Filtrations, adapted and previsible processes, stopping times. Wald's identity for sums of independent random variables.
- Lecture 26/4: Sigma-algebra of a stopping time, martingales, some properties, Doob's decomposition.
- Lecture 30/4: Martingale transform, quadratic variation, Doob's optional sampling theorem, martingale property at stopping times.
- Lecture 3/5: Asymptotics of martingales, martingale convergence theorem. Square-integrable martingales.
- Lecture 7/5: Doob's maximal and Lp inequalities.
- Lecture 10/5: Martingales closed in Lp and UI martingales. Optional stopping for closed martingales.
- Lecture 14/5: Remarks on the ABRACADABRA problem and on the Robbins-Monroe algorthm. Statement of the Martingale CLT.
- Lecture 17/5: Proof of the Martingale CLT.
- Lecture 21/5: End of the proof of the CLT. The tail sigma algebra and Kolmogorov's 0/1 law.
- Lecture 24/5: Kolmogorov's law of large numbers and Kakutani's theorem.
- Lecture 28/5: Optimal stopping problems: setting and some examples. Moser's problem and the secretary problem.
- Lecture 31/5: Value process, Snell's envelope and existence and characterisation of optimal stopping times. Solution of Moser's problem. Markovian optimal stopping problems.
- Lecture 4/6: Markov processes. Random recurrences and the Markov property. Transition kernel and initial law. Law of a Markov chain.
- Lecture 7/6: Canonical space. Strong Markov property. Martingale problems.
- Lecture 18/6: Martingale problems, relation with potential analysis, maximum principe, computations of some probabilistic quantities via equations involving the generator.
- Lecture 21/6: Discrete chains, transience and recurrence, invariant measures.
- Lecture 25/6: General stochastic processes, canonical space, product sigma-algebra, the construction of Daniell-Kolmogorov
- Lecture 28/6: Examples of stochastic processes. White noise. Markov processes in continuous time. Gaussian processes indexed by Hilbert space. Brownian motion as a Gaussian process.
- Lecture 2/7: (by F. de Vecchi) Definition of Brownian motion and equivalent characterizations as Markov process and as Gaussian process. Haar functions and Lévy contsruction of Brownian motion (first part).
- Lecture 5/7: (by F. de Vecchi) Lévy construction of Brownian motion (second part). Non-differentiability of Brownian motion. Hölder continuity of Brownian motion. Brownian motion as limit of random walks: finite dimensional marginals convergence.
- Lecture 9/7: pre-exam
- Lecture 12/7: (by F. de Vecchi) Brownian motion as limit of random walks: convergence in the space of continuous functions (Donsker theorem).