# 105.734 AKANA AKFVM AKNUM AKWTH Monte-Carlo Methods This course is in all assigned curricula part of the STEOP.\$(function(){PrimeFaces.cw("Tooltip","widget_j_id_20",{id:"j_id_20",showEffect:"fade",hideEffect:"fade",target:"isAllSteop"});});This course is in at least 1 assigned curriculum part of the STEOP.\$(function(){PrimeFaces.cw("Tooltip","widget_j_id_22",{id:"j_id_22",showEffect:"fade",hideEffect:"fade",target:"isAnySteop"});}); 2021W

2021W, VU, 3.0h, 4.5EC

## Properties

• Semester hours: 3.0
• Credits: 4.5
• Type: VU Lecture and Exercise
• Format: Distance Learning

## Learning outcomes

After successful completion of the course, students are able to...

• to explain, how Monte-Carlo methods work and which mathematical theories are behind them,
• to use Monte-Carlo methods to solve concrete problems in science, economics and finance.

## Subject of course

Monte-Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results of  problems which are difficult or impossible to solve analytically. [wikipedia

In this course we focus on the following topics:

• generating random variables
• convergence, error estimates, (law of large number, central limit theorem, concentration inequalities)
• variance reduction
• some basis result about Brownian motion, Itô's integral w.r.t. Brownian motion, Itô's formula,
• basic existence, uniqueness, stability results of stochastic differential equations (SDE), and properties of the solutions
• Feynman-Kac theorems, relation between SDEs and PDEs
• numerics of SDEs: Euler-Maruyama scheme, Milstein scheme, high-order schemes, Multi-level Monte-Carlo
• statistical error in the simulation of SDEs
• simulation of non-linear processes, backward SDEs and non-linear version of Feynman-Kac theorems
• McKean-Vlasov SDE and simulation
• stochastic optimization
• Markov chain Monte-Carlo

## Teaching methods

• Lecture online via ZOOM (or offline lecture if possible)

## Mode of examination

Immanent

The course is planned for Master’s students, but advanced Bachelor's students are also welcome. Since the topic of the course is very interdisciplinary and planned for all maths programmes, the lecturer will repeat and summarize the necessary results (from analysis, measure theory, theory of stochastic processes and numerics) at the beginning of the course.

## Course dates

DayTimeDateLocationDescription
Fri09:30 - 12:3008.10.2021 - 28.01.2022 Zoom / siehe TUWEL (LIVE).
AKANA AKFVM AKNUM AKWTH Monte-Carlo Methods - Single appointments
DayDateTimeLocationDescription
Fri08.10.202109:30 - 12:30 Zoom / siehe TUWEL.
Fri15.10.202109:30 - 12:30 Zoom / siehe TUWEL.
Fri22.10.202109:30 - 12:30 Zoom / siehe TUWEL.
Fri29.10.202109:30 - 12:30 Zoom / siehe TUWEL.
Fri05.11.202109:30 - 12:30 Zoom / siehe TUWEL.
Fri12.11.202109:30 - 12:30 Zoom / siehe TUWEL.
Fri19.11.202109:30 - 12:30 Zoom / siehe TUWEL.
Fri26.11.202109:30 - 12:30 Zoom / siehe TUWEL.
Fri03.12.202109:30 - 12:30 Zoom / siehe TUWEL.
Fri10.12.202109:30 - 12:30 Zoom / siehe TUWEL.
Fri17.12.202109:30 - 12:30 Zoom / siehe TUWEL.
Fri14.01.202209:30 - 12:30 Zoom / siehe TUWEL.
Fri21.01.202209:30 - 12:30 Zoom / siehe TUWEL.
Fri28.01.202209:30 - 12:30 Zoom / siehe TUWEL.

## Examination modalities

Active participation, exercises.
Possible to improve the grade with an oral exam.

## Course registration

Begin End Deregistration end
17.09.2021 00:00 31.10.2021 23:59 31.10.2021 23:59

## Literature

No lecture notes are available.

## Previous knowledge

• Necessary: Basic knowledge from the first four semester of the bachelors programmes, especially from measure and probability theory, statistics, as well as differential equations.
• Advantage: Basic knowledge about stochastic processes, stochastic analysis, numerics, and partial differential equations

## Language

if required in English