Lecturer
Dr. Kirsten Schorning

Teaching assistant
Dr. Vitalii Makogin

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Time and Place

Lecture
Monday, 12.30 - 2.00 pm  (220, Helmholtzstr. 18)
Wednesday,  8.30 - 10.00 am (220, Helmholtzstr. 18)

Excercise session
Friday, 8 - 10 am (120, Helmholtzstr. 18)

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Type

4 hours lecture and 2 hours excercise

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Prerequisites

Basic knowledge of probability calculus and statistics as taught, for example, in "Elementare Wahrscheinlichkeitsrechnung und Statistik".

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Intended Audience

Bachelor students in "Mathematik", "Wirtschaftsmathematik" and "Mathematische Biometrie". For master students who did not visit the course "Stochastic Stimulation" during the Bachelor and who want to attend the lecture, we offer an additional reading course.

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Contents

In this course, we focus on simulating probabilistic objects, including many important stochastic processes and structures. No prior knowledge of the probabilistic objects we study will be assumed. They will be introduced and some key properties will be examined. We also cover some basic results for measuring the accuracy of Monte Carlo estimates.

We will begin by considering random walks on graphs. We will then cover some basic theory about Markov chains that allows us to develop a number of simulation techniques. We will use these techniques to explore some spatial objects. We will look at some bounds on errors of various Monte Carlo estimators and also investigate some methods to improve efficiency when estimating various quantities related to stochastic processes and spatial objects. A number of real-world examples will be considered, mainly from physics and finance.

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Requirements and Exam

In order to participate in the final exam, it is necessary to earn 50% of the points on all theory and 50% of the points on all programming exercises.

There will be the opportunity for an oral exam at

• July 26, 2017
• July 28, 2017
• July 31, 2017
• September 27, 2017
• September 28, 2017.

For an appointement please either come to the lecture on the 5th or 7th of July or to Kirsten's office hour (every Monday 2.30-3.30 pm).

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Problem Sheets and Reading course

In order to receive points for your problem sheets, a registration at SLC is required. At the beginning of each Exercise session, students will be asked to fill the form and indicate solved exercises. Students might be asked to present their solutions. If you are not able to attend the session, please send your solution at makogin.vitalii@uni-ulm.de till 8 am before exercises.

• Exercise sheet 1 (till May 5, 2017)
• Exercise sheet 2 (till May 12, 2017)
• Exercise sheet 3 (till May 19, 2017)
• Exercise sheet 4 (till May 26, 2017)
• Exercise sheet 5 (till June 2, 2017)
• Exercise sheet 6 (till June 9, 2017)
• Exercise sheet 7 (revised) (till June 14, 2017), exercise 7-6 till June 23.
• Exercise sheet 8 (till July 12, 2017)
• Exercise sheet 9 (till July 14, 2017)

The instructions for the reading course are summarized here.

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Lecture Notes

Lecture notes will be provided in this section, roughly one week after the corresponding lectures.

lecture notes

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Literature

Asmussen, S. and P. Glynn. Stochastic Simulation. Springer, 2007.

Brémaud, P. Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues. Springer, 1999.

Dubhashi, D. P. and A. Panconesi. Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, 2009.

Glasserman, P. Monte Carlo Methods in Financial Engineering. Springer, 2004.

Graham, C. and D. Talay. Stochastic Simulation and Monte Carlo Methods: Mathematical Foundations of Stochastic Simulation. Springer, 2013.

Kroese, D. P., T. Taimre and Z. Botev. Handbook of Monte Carlo Methods. Wiley, 2011.

Levin, D. A., Y. Peres and E. L. Wilmer. Markov Chains and Mixing Times. American Mathematical Society, 2009.

Møller, J. and Waagepetersen, R. P. Statistical Inference and Simulation for Spatial Point Processes. Chapman & Hall/CRC, 2003.

Ross, S. M. Simulation, Fifth Edition. Academic Press, 2012.

Winkler, G. Image Analysis, Random Fields and Markov Chain Monte Carlo Methods: A Mathematical Introduction. Springer, 2003.