University Heidelberg | SIP RG | Lecture course Probability theory 2 (SS 2022)

Time and location of the lecture course:

Wednesday 11:15-12:45 and Friday 09:15-10:45, MΛTHEMΛTIKON, INF 205, SR B
Timely announcement about the lecture you receive on Moodle.
Please register by using Müsli to obtain further details by email.

Contact:

Lecturer: Prof. Dr. Jan JOHANNES <johannes[at]math.uni-heidelberg.de>

Assistent: Bianca Neubert <neubert[at]math.uni-heidelberg.de>

Questions, please directly by email or by using the contact form.

Language:

The lecture will be given in English if there is at least one non-German speaking participant.

Exercise group:

Please register for the exercise group of Probability theory 2 by using Müsli
to obtain further details by eMail.

We propose the following exercise group:

Day	Time	MΛTHEMΛTIKON (INF 205)	Instructor
Wednesday	14:00 - 16:00	Seminarraum 6	Emma Dingel

Questions, please directly to Emma by using Müsli.

Exercise sheets:

Please upload your solution proposals every monday before 09:00 a.m. online on Moodle.
The exercise should be handed over in fixed groups of three.
The name of the uploaded file should have the form Name1_Name2_Name3_Sheet3.pdf.
Please register by using Müsli to obtain further details by eMail.


Exercise sheet:	UE01	UE02	UE03	UE04	UE05	UE06
Handing in:	05/02/2022	05/09/2022	05/16/2022	05/23/2022	05/30/2022	06/07/2022
Solutions outline:	L01.pdf	L02.pdf	L03.pdf	L04.pdf	L05.pdf	L06.pdf

Exercise sheet:	UE07	UE08	UE09	UE10	UE11	UE12
Handing in:	06/13/2022	06/20/2022	06/27/2022	07/04/2022	07/11/2022	07/18/2022
Solutions outline:	L07.pdf	L08.pdf	L09.pdf	L10.pdf	L11.pdf	L12.pdf

Admission requirements and final grade:

The written examinartion will solely determine the final grade. You have to pass the written examination, in order to successfully attend the modul.

Admission requirements:

Accepted for the written examination is someone who either
- has achieved at least 50% of the points of the exercises and
- has actively participated during the exercises
or
- was accepted for a written examination of a previous lecture named Probability theory 2 and has not lost the examination claim.

Grading rules:

There will be two written examination (probably in the beginning and end of the lecrture free time). The first and the second exam together are counted as one examination attempt. The final mark of the modul is the mark of the first passed exam.
Students who attempt the first exam and pass with a mark better then 4.0, cannot attempt the second exam.
Only those students are allowed to attempt the second exam who failed the first exam (either with the mark 5.0 or by not attempting). It is not possible to participate in the second exam to improve the mark.

Lecture outline:

All lecture notes can be found as soon as possible after the lecture separately on here as well as combined Part A (VL01-VL07), Part B (VL08-VL16) and Part C (VL17-VL24). In the following table the individual documents are ordered by subjects.

		notes	video
Chap 1	Stochastic processes
§01	Examples	VL01
§02	Review / reminder
§03	Probability measures on Polish spaces	VL02
§04	Adapted stochastic process and stopping time	VL03
§05	Martingale theory	VL04 VL05
§06	Weak convergence	VL06 VL07
Chap 2	Stochastic differential equations
§07	Existence of Brownian motion	VL08
§08	Donsker's theorem
§09	Markov properties of the Brownian motion	VL09
§10	The Itô integral	VL10 VL11
§11	Itô processes	VL12 VL13
§12	Stochastic differential equations	VL14
§13	Martingal representation	VL15 VL16	VL16
Chap 3	Ergodic theory
§14	Stationary and ergodic processes	VL17
§15	Ergodic theorems	VL18
Chap 4	Empirical processes
§16	Empirical and partial sums processes	VL19
§17	Uniform laws of large numbers	VL20
§18	Symmetrisation
§19	Univariate exponential inequalities
§20	Set indexed empirical processes	VL21
§21	Laws of large numbers	VL22
§22	Talagrand's inequality	VL23 VL24

References:

P. Billingsley: Weak convergence of measures. (Wiley, New York, 1968).

Y. S. Chow and M. Teicher: Probability Theory: Independence, Interchangeability, Martingales. (Springer-Verlag, 1987)

K. L. Chung: A Course in Probability Theory. (Harcourt, Brace & World Inc., 1968)

R. Durrett: Probability: Theory and Examples. (Cambridge University Press, Cambridge, 2010)

S. Karlin and H.Taylor: A First/Second Course in Stochastic Processes. (Academic Press, San Diego, California, 2005)

O. Kallenberg: Foundations of Modern Probability. (Springer, Berlin, Heidelberg,2002)

I. Karatzas and S. Shreve: Brownian Motion and Stochastic Calculus. (Springer, Berlin, Heidelberg, 1998)

A. Klenke: Probability Theory. A Comprehensive Course. (Springer, Berlin, Heidelberg, 2008)

J. Neveu: Martingales à temps discret. (Masson, 1972)

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