Stochastics and Probability

Stochastic Modeling and Simulation

Upon completing the module, the students master the basics of stochastic modelling and simulation. We first discuss discrete-time models, followed by two classic examples, and then continuous-time models.


Conditional probabilities, normal distributions, and scale-free distributions; Markov chains and their matrix representation, mixing times and Perron-Frobenius theory; Applications of Markov chains, such as the PageRank algorithm; Monte Carlo Methods: Convergence, Law of Large Numbers, Variance Reduction, Importance Sampling, Markov Chains Monte-Carlo Using Metropolis-Hastings & Gibbs Samplers; Random processes and Brownian motion: properties in 2, 3 and more dimensions, connection to the diffusion equation, Levy processes and anomalous diffusion; Stochastic differential equations (SDEs): Nonlinear transformations of Brownian motion (Ito calculus), Ornstein-Uhlenbeck process and other solvable equations; Examples from population dynamics, genetics, protein kinetics, etc.; Numerical simulation of SDEs: strong and weak error, Euler-Maruyama scheme, Milstein scheme.

Program / Module

M.Sc. Computational Modeling and Simulation
Module: CMS-COR-SAP - Stochastics and Probability

Winter Term

Lecture: Mondays, 4. DS (13:00-14:30) in HSZ-401 (Hörsaalzentrum) / FIRST LECTURE: OCT 14
Exercises / Tutorials: Thursday, 5.DS (14:50 - 16:20) in GÖR-229 (Görges-Bau)


2 SWS lecture, 1 SWS exercise, 1 SWS tutorial, self-study

5 credits


Wednesday, February 19, 2020, 09:20-10:50h, TRE/MATH/H (Trefftz-Bau, Haus A)

Since there are more than 10 registered students, module examination consists of a written examination with a duration of 90 minutes.

At the exam, the following may be used:

  • 4 A4 sheets (8 pages if you print duplex) of hand-written summary. We recommend writing the summary by hand, but it can also be machine-written. In the latter case, the font size must be 8 points or larger throughout.
  • A standard pocket calculator (devices with network or bluetooth access, as well as devices capable of storing and displaying documents are not allowed)
Items not adhering to these guidelines will be confiscated in their entirety at the beginning of the exam.

Exam Review

You can come and look at your exam, and ask questions about its correction and the answers given during the exam review times. Due to the current travel and contact restrictions, we offer three exam review dates: one in June, one during the summer break, and one around the beginning of the winter semester 2020/21. The dates and modalities for the latter two are yet to be determined, but you are invited to participate in the June date if you happen to be around:

  • June 18, 2pm, Outdoors at the pond between the BAR building and the Mensa (Maps Link). The review only takes place if there is no rain. In case of rain, a new date will be found. In order to participate, You MUST wear a face mask and you are only allowed to come forward one by one.
  • July 20, 2pm. Location to be announced, depending on how the Covid restrictions develop
  • October 19, 2pm. Location to be announced, depending on how the Covid restrictions develop
IMPORTANT: All students attending an exam review must fill in and sign the exam review form they are going to receive during the review. Undocumented exam reviews are not permitted.

Registration to the course

For students of the Master program "Computational Modeling and Simulation: via CampusNet SELMA

For students of the Computer Science programs: via jExam


Lecture: Prof. Ivo F. Sbalzarini & Dr. Christoph Zechner
Exercises: David Gonzales

Teaching language: ENGLISH

Lecture notes are available as PDF here.
Below is the weekly syllabus and the exercise/solution handouts:

Suggested Literature

Feller - an introduction to probability theory and its applications, Wiley+Sons, 1957.
Robert & Casella - Monte Carlo statistical methods, Springer, 2004.