# MAT-55216 Topics in Applied Mathematics, 3 credits (2012)

# Project Work

## General instructions

Each student should select a *project work topic* from the list
below and report it to the teacher latest on *November 30th*. You
can ask for more information about the topics from the lecturer. There
might also need a possibility to arrange a discussion on that week (to
be confirmed). Also note the the last topic is "Own Topic" and the
best project work would be one where you apply the methods to an
application within your own research area.

- The project work
*should be returned*to the lecturer (simo.sarkka@aalto.fi) by e-mail in PDF form latest on*December 31st*. - The document should contain:
- Introduction, which explains the research problem in informal terms.
- Theory section which describes the theory behind the application/method and cites the relevant books and scientific articles, where the theory can be found.
- Simulation/results section, where the method is applied to a simulated or real application.
- Summary section, which summarizes the results.

## Topics

Potential topics include, but are not limited to:

- Stochastic Runge-Kutta methods
- Strong and weak convergence of numerical methods
- Numerical simulation of SDEs
- Gaussian approximations of SDEs
- Variational Bayes approximations of SDEs
- Series expansions of SDEs
- Small noise expansion approximations
- Numerical solution of Fokker-Planck-Kolmogorov equations
- Solution of PDEs with Feynman-Kac
- Wiener/Feynman path integrals and SDEs
- Existence and uniqueness of SDEs
- Martingale representation theorem
- Analytical solutions of SDEs
- Kalman-Bucy filter
- Kushner and Zakai equations
- Continuous-time filtering theory
- Continuous-discrete filtering theory
- Levy-process driven SDEs
- Spatially distributed systems
- Black-Scholes formula
- Physics application
- Biological application
- Communications application
- Financial application
- Other application
- Own topic