# MAT-55216 Topics in Applied Mathematics, 3 credits

**Fall 2012:**
Applied Stochastic Differential Equations

**Teacher:**
Dr. Tech. Simo Särkkä,
Aalto University, Finland

**Coordinator:**
Prof. Robert Piché,
Mathematics Dept, TUT

**Schedule:**
Lectures &
Exercises Thursdays 1-4pm (25.10, 1.11, 8.11., 15.11., 22.11.). The
exercise hour is 1-2pm and the lecture is 2-4pm.
Location: Tampere University of
Technology room Td308 (bring a laptop with Matlab/Octave)

**Topics:**
An introduction to the theory, applications and numerical methods
for SDEs. Applications include biology, finance, computational
statistics, communications engineering, fluid dynamics, and
mechanical vibrations. After the course the student should be
able to formulate a simple SDE model for an application, analyze
its properties, and solve it numerically using appropriate
methods.

**Prerequisites:**
Multivariate differential and integral
calculus, matrix analysis, basic probability, Matlab/Octave.

**Course Pass Requirements:**

- The grade will be based on the project work. See Project work instructions and
topics. (Deadlines:
**30.11. and 31.12.**) - Total 2/3 of the home work problems must be completed (see instructions below).

## Lectures:

The slides are copied here after each lecture (the topics are preliminary and might change):

**Thursday 25.10. 2-4pm:***Pragmatic Introduction to Stochastic Differential Equations*(Slides as PDF)**Thursday 1.11. 2-4pm:***Ito Calculus and Stochastic Differential Equations*(Slides as PDF)**Thursday 8.11. 2-4pm:***Probability Distributions and Statistics of SDEs*(Slides as PDF)**Thursday 15.11. 2-4pm:***Numerical solutions of SDEs, Ito-Taylor Series, Gaussian Process Approximations*(Slides as PDF)**Thursday 22.11. 2-4pm:***Further Topics; Series Expansions, Feynman-Kac, Girsanov Theorem, Filtering Theory*(Slides as PDF)

## Exercise hours:

Before each lecture there is an exercise hour, which works as follows: In the beginning, you should flag to a list which of the exercises you have completed. One person will be randomly selected to present his/her answer on the drawing board or computer. Note that in the end of the course you must have done and flagged total of 2/3 of all the exercises during the course (8 out of 12 exercises).

On the first day 25.10. 1-2pm there will be a voluntary introductory demonstration session on ordinary differential equations before the lecture starts at 2pm.

**Thursday 25.10. 1-2pm:**This is a demonstration session based on the Chapter 1 of lecture notes.**Thursday 1.11. 1-2pm:**Exercise Round 1**Thursday 8.11. 1-2pm:**Exercise Round 2**Thursday 15.11. 1-2pm:**Exercise Round 3**Thursday 22.11. 1-2pm:**Exercise Round 4

**Attention:** Below you can find a set of extra exercises which can be returned to reach the required amount of
8 points. You can get single point from each exercise and the answers should be
returned to the lecturer in PDF/Matlab form by **December 16th, 2012**:

**DL 16.12.2012:**Extra Exercise Round 5

## Project Work:

The grade of the course will be based on a project work. See Project work instructions and
topics. (Deadlines: **30.11. and 31.12.**)

## Course Material:

The primary course material consists of a set of lecture notes
and slides which are available here in PDF form. *The lecture notes
can be found below*:

Students might find the following books useful as well:

- Gardiner: Stochastic Methods: A Handbook for the Natural and Social Sciences, Springer.
- Gardiner: Handbook of Stochastic Methods, Springer (old version of the above).
- Oksendal: Stochastic Differential Equations, Springer.