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.

Topic Prerequisites

Working knowledge of computer programming in any language (e.g. Matlab, Python, Java), basic knowledge of classical physics, and solid undergraduate knowledge in calculus, probabilities and statistics.

Program / Module

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

Format

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

Registration to the course

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

Instruction language: ENGLISH

Script

Lecture notes are available as PDF here.

Suggested Literature

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