## 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.

**Contents**

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

**Time/Place**

**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)

**Format**

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

5 credits

**Exam**

**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)

**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

Teachers

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, Solution 08 PDF, Solution Jupyter Notebook)
- Lecture 9 (Sbalzarini) - Monte-Carlo optimization: stochastic gradient descent, simulated annealing, evolution strategies, CMA-ES (Exercise 09 PDF, Solution 09 PDF, Solution Jupyter Notebook)
- Lecture 10 (Zechner) - Random Walks, Brownian motion in 1,2,3,n-dim, connection to diffusion, continuum limit of random walks (Exercise 10 PDF, Solution 10 PDF, Solution Jupyter Notebook)
- Lecture 11 (Zechner) - Stochastic calculus, Ito calculus, Ornstein-Uhlenbeck process (analytical) (Exercise 11 PDF, Solution 11 PDF, Solution Jupyter Notebook)
- Lecture 12 (Sbalzarini) - numerical methods for SDE: Euler-Maruyama, Milstein, strong and weak convergence (Exercise 12 PDF, Solution 12 PDF, Solution Jupyter Notebook)
- Lecture 13 (Zechner) - Master equation, Fokker-Planck, Kolmogorov forward, Example: chemical kinetics (Exercise 13 PDF, Solution 13 PDF)
- Lecture 14 (Sbalzarini) - Graphical representation and classification of reaction networks, exact stochastic simulation algorithms: first-reaction method, direct method (Exercise 14 PDF, Solution 14 PDF, Solution Jupyter Notebook)

**Suggested Literature**

Feller - an introduction to probability theory and its applications, Wiley+Sons, 1957.

Robert & Casella - Monte Carlo statistical methods, Springer, 2004.