The project

Introduction

  • This project is compulsory: To take the exam, you must prepare a small report about a topic related to stochastic differential equations not covered in the lectures.

  • You may study any relevant topic of you own choosing, see below for suggestions and more information.

  • There will be a 2 week break in the lectures to do the project.

General information

  • The report should:
    • be a 4-6 pages discussion of a limited topic
    • be typed using with a 10-12 points font size
    • be in English (or Norwegian)
    • use notation and defnitions consistent with Øksendal and what we use in the course
    • be on a level suitable for fellow students with a comparable background.
      The presentation should be informative, professional, and not too formal.
  • Suggested layout:
    • A title and name
    • A short abstract
    • A short general introduction describing the problem
    • The main body. Lengthy mathematical proofs (unless original!) should not be included, but precise references should be given.
    • A list of references, all of which should be used in the text
    • Avoid appendices as far as possible, except for computer code and possible figures.
  • Deadlines:
    • October 13: email me your choice of project.
    • December 1st: 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

  1. 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).

  2. The general Ito integral and formula.

    Extension of the \(L^2\)-theory using stopping-times and convergence in probability. Start by reading chapter 15 and 16 in Schilling and/or chapter 5 in Kuo.

  3. 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.

  4. 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.

  5. Levy Processes

    The Levy processes are natural generalizations of Brownian motions. They share properties like iid increments, but this class of stochastic processes is much wider and includes in addition to Brownian motions also processes which may have jumps. An Ito like theory for stochastic integrals and SDEs exists. The book Levy Processes and Stochastic Calculus by D. Applebaum has a lot of information on this subject. See also the book by Kuo chp. 6.

  6. 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.

    Sample path simulations, e.g. Euler-Maruyama and Milstein methods, strong and weak methods (and convergence). The book by Kloeden and Platen: Numerical Solutions of Stochastic Differential Equations is a classic (start with chp. 9). For SDEs with jumps, see e.g. the book by Platen and Bruti-Liberati.

    Computations of densities. Sufficient in many applications and often very much faster than sample path simulations. See e.g. On numerical density approximations of solutions of SDEs with unbounded coefficients and references therein.

  7. Backward stochastic differential equations.

    Martingale representation theorem, additional unknown process to get adapted solutions (1 equation, 2 unknowns!). There are several books and lecture notes on the topic, e.g. by Carmona and Pardoux.

  8. Space-time white noise and SDEs in Hilbert spaces.

    Technical and ambitious, the basis for doing stochastic PDEs (SPDEs). Explain some simple explicit examples. Advice: Do not go all the way to SPDEs. See e.g. chp 2 and 3 in the lecture note of Kovacs and Larsson. See also this page for more literature.

  9. 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 floating 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.

  10. 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.
2023-10-05, Espen Robstad Jakobsen