# Stochastic Simulation

Lecturer
Dr. Vitalii Makogin

Teaching assistant
Duc Nguyen

## Time and Place

Period of the lectures
from 19.04.2021 to 16.07.2021
Lecture
Will be published online weekly on Wednesday at 10.00

Course page on moodle

Exercise session
Online session per moodle each 2 weeks on Thursday from 12:00 to 14:00

## Prerequisites

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

## 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.

## Contents

In this course, we focus on simulating of probabilistic objects, including many important stochastic processes and structures (Markov chains, etc.) as well as a plentiful number of methods and advanced knowledge about stochastic processes. No prior knowledge of the probabilistic objects we study will be assumed. Each topic will be provided with several practical examples.

First part of the lectures is about the concept of discrete-time Markov chains, its different representation and properties, especially the (strong) Markov property. Once we have such representation, we will be able to simulate general discrete-time Markov chain and Random walk on graphs as a special case. In addition, to understand the mechanism, reversibility and stopping times will be investigated. In the second part, one can learn how to draw samples of random elements. This procedure provides some crucial methods such as inversion method, acceptance-rejection algorithm, Metropolis and Metropolis-Hastings algorithms, which belong to the class of algorithms so-called Markov chain Monte Carlo(MCMC). We discuss their initial conditions and study the convergence property which uses the total variation distance. In the classical law of large numbers, the sample mean converges to theoretical mean under the independence and identical distribution of entries. In this course, we will simulate a discrete Markov chain and prove that the sample mean also converges to the theoretical mean and the Strong Law of Large Number holds.

We extend an index set to a multi-dimensional space and cover Markov and Gibb’s random fields. Eventually, we come to the Markov chains defined over general space and some similar to one-dimensional results will be provided.

## 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.

## Problem Sheets and Reading course

Problem sheets will be provided once per two weeks. The deadline for the submission is two weeks after the exercises are published. Students might be asked to present (or explain) their solutions as well as answer some additional questions.

## Lecture Notes

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

## 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.

# Contact

## Lecturer

Dr. Vitalii Makogin

Office hour: on appointment via e-mail

Telefon: +49 (0)731/50-23527

Homepage

## Teaching assistant

Duc Nguyen

Telefon: +49 (0)731 / 50 - 23529

Email: tran-1.nguyen@uni-ulm.de