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 20, 2019, 09:20-10:50h, CHE/089/E (chemistry building)

If there are more than 10 registered students, the module examination consists of a written examination, with a duration of 90 minutes. If there are 10 or fewer registered students, it consists of an oral examination as an individual examination performance amounting to 30 minutes; this will be announced to the enrolled students at the end of the enrollment period.

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 following times:

  • April 15: 14:00-15:00
  • April 16: 15:00-16:00
  • April 17: 9:00-10:00

All exam check session are going to happen at the CSBD (Pfotenhauerstr. 108) in the rooms of the Professorship.

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:

  • Lecture 1 (Sbalzarini) - Probability refresher, conditional probabilities, Bayes' rule, random variables, discrete and continuous probability distributions, scale-free distributions (Exercise 01 PDF, Solution 01 PDF, Solution Jupyter Notebook)
  • Lecture 2 (Sbalzarini) - transformation of random variables, pseudo- and quasi-random numbers, low discrepancy sequences, transformation algorithms: inversion, Box-Muller, accept-reject method, composition-rejection method (Exercise 02 PDF, Solution 02 PDF, Solution Jupyter Notebook)
  • Lecture 3 (Sbalzarini) - Discrete-time stochastic processes, discrete Markov chains and their matrix (Exercise 03 PDF, Solution 03 PDF, Solution Jupyter Notebook)
  • Lecture 4 (Sbalzarini) - Law of large numbers, Monte Carlo methods, example: MC integration, importance sampling (Exercise 04 PDF, Solution 04 PDF, Solution Jupyter Notebook)
  • Lecture 5 (Zechner) - Monitoring variance, variance reduction (Exercise 05 PDF, Solution 05 PDF, Solution Jupyter Notebook)
  • Lecture 6 (Zechner) - Rao-Blackwell, Markov Chain Monte Carlo (MCMC), detailed balance, convergence criteria, acceleration methods (Exercise 06 PDF, Solution 06 PDF, Solution Jupyter Notebook)
  • Lecture 7 (Zechner) - Classic MCMC samplers 1: Gibbs sampling (Exercise 07 PDF, Solution 07 PDF, Solution Jupyter Notebook)
  • Lecture 8 (Zechner) - Classic MCMC samplers 2: Metropolis-Hastings, convergence dagnostics, stopping conditions (Exercise 08 PDF)
  • Lecture 9 (Sbalzarini) - Monte-Carlo optimization: stochastic gradient descent, simulated annealing, evolution strategies, CMA-ES
  • Lecture 10 (Zechner) - Random Walks, Brownian motion in 1,2,3,n-dim, connection to diffusion, continuum limit of random walks
  • Lecture 11 (Zechner) - Stochastic calculus, Ito calculus, Ornstein-Uhlenbeck process (analytical)
  • Lecture 12 (Sbalzarini) - numerical methods for SDE: Euler-Maruyama, Milstein, strong and weak convergence
  • Lecture 13 (Zechner) - Master equation, Fokker-Planck, Kolmogorov forward, Example: chemical kinetics
  • Lecture 14 (Sbalzarini) - Graphical representation and classification of reaction networks, exact stochastic simulation algorithms: first-reaction method, direct method.
Suggested Literature

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