|Abstract||This course introduces statistical Monte Carlo methods. This includes applications of stochastic simulation in various fields (statistics, statistical mechanics, operations research, financial mathematics), generating uniform and arbitrary random variables (incl. rejection and importance sampling), the accuracy of methods, variance reduction, quasi-Monte Carlo, and Markov chain Monte Carlo.|
|Objective||Students know the stochastic simulation methods introduced in this course. Students understand and can explain these methods, show how they are related to each other, know their weaknesses and strengths, apply them in practice, and proof key results.|
|Content||Examples of simulations in different fields (statistics, statistical mechanics, operations research, financial mathematics). Generation of uniform random variables. Generation of random variables with arbitrary distributions (including rejection sampling and importance sampling), simulation of multivariate normal variables and stochastic differential equations. The accuracy of Monte Carlo methods. Methods for variance reduction and quasi-Monte Carlo. Introduction to Markov chains and Markov chain Monte Carlo (Metropolis-Hastings, Gibbs sampler, Hamiltonian Monte Carlo, reversible jump MCMC). Algorithms introduced in the course are illustrated with the statistical software R.|
|Lecture notes||A script will be available in English.|
|Literature||P. Glasserman, Monte Carlo Methods in Financial Engineering.|
B. D. Ripley. Stochastic Simulation. Wiley, 1987.
Ch. Robert, G. Casella. Monte Carlo Statistical Methods.
Springer 2004 (2nd edition).
|Prerequisites / Notice||It is assumed that students have had an introduction to probability theory and statistics (random variables, joint and conditional distributions, law of large numbers, central limit theorem, basics of measure theory).|
The course resources (including script, slides, exercises) will be provided via the Moodle online learning platform.