(7) Mathematics, Statistics, or Computer Science (from IMADA)
These topics are generally only applicable for scient.oecon. students.
Topics from the above subject areas (when admissible) can also be chosen by scient.oecon. students. If an IMADA supervisor desired, chose Mathematics, Statistics, or Computer Science as your main topic area and refer to the topic number in the selected topic application.
Questions about the topics within the subject area of
- Mathematics: Kristian Debrabant, e-mail: email@example.com
- Statistics and probability theory: Kristian Debrabant, e-mail: firstname.lastname@example.org
- Numerical analysis/Computer science: Kristian Debrabant, e-mail: email@example.com
All topics in this subject area can be written in English or Danish.
7.1.a Surrogate modeling of option prize values via Gaussian process regression
In its general form for multiple assets, there is no known closed-form solution to the Black-Scholes equations. Moreover, numerical approximations may quickly become very large due to the curse of dimensionality. The objective of this thesis is to construct and examine surrogate models for the Black-Scholes equation under suitable parametric variations via using Gaussian process regression. Gaussian process regression can be thought of as a statistical multivariate interpolation method that allows incorporating gradient information as well as data layers of variable fidelity.
Santner, T.J. and Williams, B.J. and Notz, W.~I.: The Design and Analysis of Computer Experiments, Springer, New York Berlin Heidelberg, 2003
Espen Benth, F.: Option Theory with Stochastic Analysis: An Introduction to Mathematical Finance, Springer, Berlin, Heidelberg, 2004
Prerequisites: MM533, ST521 (IMADA)
(This topic is related to finance; in particular the topic derivatives)
7.2.a Network dynamics
Networks such as the Internet, the World Wide Web, and social and biological networks permeate our modern societies. A lot of recent study on networks takes a dynamical systems view according to which the vertices of a graph represent discrete dynamical entities, with their own rules of behavior, and the edges represent direct interactions between the entities. Such networks, for instance epidemic network in which a virus is spreading or economic networks which host the capital flows, have not only topological properties but have dynamical properties as well.
The aim of this project is to introduce the student to concepts and methods for the analysis of networks and their dynamical properties. The topics of this project cover the following
- Graph theory: basic concepts, theorems and algorithms regarding network flows;
- Markov Chains: basic concepts and results;
- Linear dynamic systems on networks;
- Random graphs: basic concepts and theorems;
- Application of network dynamics in Epidemics or economics.
R. Diestel, Graph Theory, 4th edition.
M. Newman, A. Barabasi and D. J. Watts, The structure and Dynamics of Networks.
O. Haeggstroem, Finite Markov Chains and Algorithmic Applications.
S. I. Resnick, Adventures in stochastic processes.
7.3.a Computational Optimization
Projects in computational optimization deal with cases similar to those described in chapter “Optimization and Logistics” but focus on mathematical modeling, implementation of the model in a programming language learned during the bachelor, experimentation, analysis of results and technical writing. If existing solvers are not enough to solve the problem at hand, then the implementation part will expand to develop advanced, ad-hoc techniques, as it is often the case for difficult optimization problems. Examples of advanced techniques are: column generation, branch and cut, metaheuristics.
Specific industrial cases may arise in connection with an industrial partner at the time of applying for the bachelor. They can also be brought by the student.
Recent cases included the optimization of:
- Production scheduling in manufacturing
- Routing in distribution and services
- Course/exam timetabling
- Train timetabling
- Revenue management in the airways sector
Prerequisites for these projects are the courses:
- DM545 Integer and Linear Programming
- DM550 Introduction to Programming
- DM507 Algorithms and Data structures
- Credit risk analysis: statistical models to predict the defaults in credit requests
- Computational social choice: how hard is it to determine the winner of a certain election system or how traditional optimization problems can be formulated with an objective function that takes into account not only a utilitarian but also a social welfare point of view.
- Crowd intelligence: using the opinions of experts to improve performance in automatic prediction systems.
- Visualization methods: how to efficiently and effectively visualize data such that we can make sense of them.
7.4.a. Extreme value statistics and reinsurance pricing
Extreme value statistics deals with modelling extreme events, that is events with a low frequency of occurrence but a high impact. At the theoretical level this means that we focus on studying tails of distribution functions, while for practical data analysis the focus will be at the largest observations in a random sample. Insurance claim data typically exhibit a heavy-tailed behaviour, which makes the class Pareto-type distributions a good candidate for modelling this type of data. In the project it is the intention to study this class of heavy tailed distributions and apply it to a dataset of car insurance claims. Insurance companies typically protect themselves against large claims by entering into re-insurance contracts with re-insurance companies. In such contracts the re-insurer intervenes if a claim exceeds the level specified in the re-insurance contract and pays the excess over that level. Re-insurers only intervene for extreme claims and hence extreme value statistics is a crucial tool for them, in order to obtain an accurate description of the upper tail of the claim size distribution. In the project focus will also be on pricing re-insurance contracts based on extreme value statistics.
7.4.b Martingale convergence
A martingale is an elegant and powerful mathematical model that can be considered as a generalisation of the well-known sums of independent random variables. Martingales play an important role in probability theory and statistics. Besides they are extremely applicable and form the basis for a lot of models in mathematical finance. With the project it is the intention to study the main martingale convergence results, in particular (i) the martingale convergence theorem and its refinements under uniform integrability, (ii) a proof of the strong law of large numbers for independent and identically distributed sequences of random variables using backward martingales and (iii) a martingale central limit theorem.
Gut, A., 2013. Probability: a graduate course. Springer.
Jacod, J., Protter, P., 2004. Probability essentials. Springer.
Schilling, R.L., 2005. Measures, integrals and martingales. Cambridge University Press.