Course Syllabus for

# Numerical Methods for Deterministic and Stochastic Differential Equations Beräkningsmetoder för deterministiska och stokastiska differentialekvationer

## FMN010F, 7.5 credits

Valid from: Autumn 2013
Decided by: FN1/Anders Gustafsson
Date of establishment: 2014-09-15

## General Information

Division: Numerical Analysis
Course type: Third-cycle course
Teaching language: English

## Aim

Stochastic differential equations are increasingly important in many cutting-edge models in physics, biochemistry and finance. The aim of the course is to give the postgraduate student a fundamental knowledge and understanding of stochastic differential equations, emphasizing the computational techniques necessary for stochastic simulation in modern applications.

## Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• have a good understanding of the difference between stochastic and deterministic equations, and of Monte Carlo methods for stochastic simulation.
• be able to analyse basic methods for stochastic differential equations, such as the Euler-Maruama and Milstein methods, and more general Runga-Kutta methods.
• have a good understanding of weak and strong convergence, and of stability theory for stochastic differential equations.
• be able to interpret stochastic differential equations and to give examples of models that include them.

Competences and Skills

For a passing grade the doctoral student must be able to independently implement discretization schemes for stochastic differential equations and critically evaluate the results.

## Course Contents

The course is divided into two parts, with the first dealing with classical theory for deterministic ordinary differential equations (ODEs) and the second with theory for stochastic differential equations (SDEs). The deterministic part reviews necessary background, in particular Runge-Kutta and Rosenbrock methods. The second part gives an introduction to SDEs, and presents basic ideas and techniques used in statistical simulation, such as root mean square stability; consistency notions; and weak and strong convergence. A few applications will be studied in more detail, including pertinent problems to be solved using a computer.

## Course Literature

Averina, Tatjana A.: Numerical analysis of systems of ordinary and stochastic differential equations. V.S.P. International Science, 1997. ISBN 9789067642507.

## Instruction Details

Types of instruction: Lectures, project

## Examination Details

Examination formats: Oral exam, seminars given by participants
Examiner: