The project

Introduction

To take the exam you must prepare a small report about a topic related to stochastic differential equations not covered in the lectures.
The project counts for 20% of the final grade:
- Students taking MA8109: The project is graded based on the report.
- Students taking TMA4505: The project is graded based on a 10 minutes presentation of the report during the oral exam.
You may study any relevant topic of you own choosing, see below for suggestions and more information.

General information

Each student delivers one written report.
- Students taking TMA4505 should also give a 10 min informal presentation at the oral exam.

The report should:
1. be a 4-6 pages discussion of a limited topic
2. be typed using with a 10-12 points font size
3. be in English (or Norwegian)
4. use notation and defnitions consistent with Øksendal and what we use in the course
5. be on a level suitable for fellow students with a comparable background.
  The presentation should be informative, professional, and not too formal.

Suggested layout:
1. A title, student number, and candidate number (no names)
2. A short abstract
3. A short general introduction describing the problem
4. The main body. Lengthy mathematical proofs (unless original!) should not be included, but precise references should be given.
5. A list of references, all of which should be used in the text
6. Avoid appendices as far as possible, except for computer code and possible figures.

Deadlines:
1. October 18: email me your choice of project.
2. November 29: email me the written report.

OBS:
- You are welcome to find and use additional literature, but be careful with material found on the Internet.
- Students are free to cooperate, but are expected to be honest and open about that.
- Because of the time and size limitations, it is important to keep a limited scope for the work!

Project Proposals

The Stratonovich Integral

The Stratonovich stochastic integral is an alternative to the Ito integral. Write a short essay about the Stratonovich integral based on Øksendal and other relevant literature. Use our notation and review main properties and important results (Obviously, there are advanced results not so easily available. Do not spend time on those).
The Kalman-Buzy Filter

The Kalman-Buzy filter is described in Chapter 6 in Øksendal and is a very important application of stochastic differential equations. The chapter contains several examples, and 2-3 students could actually study this chapter and select different examples for their main study. There are also numerous other references available for this topic.
The Black-Scholes Model and the Pricing of Options

The Black-Scholes model is the most famous application of stochastic differential equations to Financial mathematics. The model is described in Chapter 12 in Øksendal and in numerous other places. Also this project can easily accommodate more than one student. It may be worthwhile to look for some simpler treatments. The book of Thomas Mikosch is one option. Norwegian students could see the book Matematisk Finans by Fred Espen Benth. See also the discussion in Øksendal and Evans.
Levy Processes

Brownian motion is a simple example of a Levy Process, but this class of stochastic processes is much wider. A characteristic feature is that the paths may have jumps. This project may be a bit hard without any prior knowledge.
Complex Brownian Motion

This project requires some prior investigations and may not be very suitable, since my own knowledge of complex Brownian motion amounts to Exercise 5.14 in Øksendal. The projectclearly requires some knowledge of complex analysis.
Numerical Solutions of Stochastic Differential Equations.

This is a wide field suitable for students with some programming and numerics background. The project could study some of the suggested numerical methods and their behaviour for equations with known, explicit solutions.
The book by Kloeden and Platen: Numerical Solutions of Stochastic Differential Equations is a classic in this field.
Projects Based on Exercises in Øksendal.

Some exercises in Chapter 5 could be expanded into an essay:
- Exercise 5.12: Linear pendulum subject to random perturbations
- Exercise 5.13: Slow drift of ships and áoating platforms subject to random wave forcing.
- Exercise 5.15: Population growth in a stochastic, crowded environment. There are various other related models that could also be studied.
- Exercise 5.16: Use of integrating factors for solving various SDEs.
- Exercise 5.18: The geometric mean reversion process (see also 5.7) and applications.
Open Projects

All students are free to find other topics which they have found interesting, but we should then have a discussion before starting serious work.