19212901 Basismodul: Stochastics II
Summer Term 2020
lecture Prof. Dr. Nicolas Perkowski, exercise joint with Helena Kremp
Time and place

Lecture: Video lectures will be available online (see below).
Online "office hours": Tuesdays 12:3014:00. The link to the Webex session can be found in the Whiteboard system.  Exercise Session: Tuesdays, 16:0017:30, online. The link to the Webex session can be found in the Whiteboard system.
 Final Exam: to be announced in due course
Prerequisits: Stochastics I und Analysis I — III.
Assessment
To receive credits fo the course you need to
 actively participate in the exercise session
 work on and successfully solve the weekly exercises
 pass the final exam (see above)
If you are an FU student you only need to register for the course via CM (Campus Management).
If you are not an FU student, you are required to register via KVV/Whiteboard.
Exercises
Problem sets will be put online every Tuesday and can be found under Assignements in the KVV/Whiteboard portal. Solutions (in groups of three!) are due by 4pm on Tuesday of the following week – solutions must be uploaded through the KVV/Whiteboard portal or sent by email to Helena Kremp.
Course Overview/ Content:
 Construction of stochastic processes;
 conditional expectation;
 martingales in discrete time: convergence, stopping theorems, inequalities;
 convergence types in stochastics;
 uniform integrability;
 Markov chains in discrete and continuous time: recurrence and transience, invariant measures;
 convergence in distribution of stochatic processes;
 Brownian motion and invariance principle.
References
 Klenke: Wahrscheinlichkeitstheorie
 Durrett: Probability. Theory and Examples.
Further literature will be given during the lecture.
Lecture videos
 Videos for Tuesday, April 21:
 Welcome
 Introduction (Section 0 of the notes)
 Definition of stochastic process (1.11.2)
 Distribution of a stochastic process (1.31.5)
 Finitedimensional distributions of a stochastic process (1.61.7)
 Videos for Thursday, April 23:
 Definition and Examples of Polish spaces (1.81.10)
 Tightness and regularity of probability measures (1.111.12)
 Probabilites on Polish spaces are tight (1.131.15)
 Videos for Tuesday, April 28:
 Videos for Thursday, April 30:
 Construction of product measures & Gaussian processes (1.191.20)
 First example of a Markov chain (1.21)
 Definition of a Markov chain (1.221.26)
 Construction of Markov chains (1.271.28)
 Videos for Tuesday, May 5:
 Videos for Thursday, May 7:
 (Short) Videos for Tuesday, May 12:
 Videos for Thursday, May 14:
 Filtrations and martingales (3.13.3)
 Examples of martingales (3.53.6)
 Martingale transforms (3.73.8)
 Convex functions of martingales (3.113.12)
 Videos for Tuesday, May 19:
 Stopping times (3.133.15)
 Stopping theorem (3.163.20)
 Events observable until a stopping time (3.213.22)
 Optional sampling theorem (3.243.26)
 Videos for Tuesday, May 26:
 Applications of the optional sampling theorem (3.273.28)
 Doob's upcrossing inequality (3.293.32)
 Martingale convergence theorem (3.29)