Stochastics II

Prof. Dr. Henryk Zähle

Tristan Bergmann, MSc.

Winter Term 2020/2021

 

Due to the COVID-19 pandemic this lecture course will be held online. Detailed lecture notes will be updated weekly and discussed in a MS Teams session once a week (Thursday, 10-12). The lecture notes as well as the exercise sheets will be available on Moodle.

The Moodle platform can be accessed with the UdS username and password. For details on Microsoft Teams please see Microsoft Teams.

Exercises

As usual, there will be an exercise sheet every week. To be admitted to the exam, you must achieve a certain score to be fixed later. As mentioned above, the exercise sheets will be available on Moodle. The solutions will be discussed every week in a tutorial (in the form of a MS Teams meeting).

Contents

•    L2 as Hilbert space, orthogonal projections
•    conditioning on sigma-algebras, in particular conditional distributions and expectations
•    foundations of general stochastic processes
•    Poisson process
•    Gaussian processes
•    Brownian motion
•    Markov processes
•    Martingales

Prerequisites for attending

For attending this course it is required is that you have successfully attended the lecture series Analysis I-III, Linear Algebra I-II as well as the lecture course Stochastics I (or equivalent courses). The content of the course Stochastics I is:

•    profound measure and integration theory; in particular Lp theory, limit theorems, Radon-Nikodym derivative, product measures
•    general probability spaces
•    random variable and their distributions
•    conditioning on events
•    the concept of independence
•    expectation, variance, covariance, correlation, higher moments
•    characterization of distributions on Euclidean space; in particular distribution functions, generating functions, characteristic functions
•    sums of independent random variables; convolutions
•    convergence of sequences of probability measures and random variables
•    limit theorems for real random variables (SLLNs, CLT)
•    multivariate normal distribtion, multivariate CLT

Exam

The oral exams will be held in February and March 2021. More information will be announced in time.

Related lecture courses in the next two semesters

Mathematical Statistics, 4h (Sommer Term 2021)

Time Series Analysis, 2h or 4h (Winter Term 2021/22)