Student Projects

Supervisor: Jochen Voss — J.Voss@leeds.ac.uk

General Remarks

I am happy to supervise student projects and assignments. If you want to do a project with me, here is some advice:

• I am interested in the broad areas of probability and computational mathematics. This includes for example stochastic processes, Monte Carlo methods, random number generation and stochastic analysis.
• I do not know much about financial mathematics.
• For projects with a computational component, I am happy for you to use R, Python, Matlab or C/C++ to implement programs.
• You are welcome to suggest your own topic; just get in touch and discuss your plans/wishes with me. If you can't think of a project on your own, a list of suggested projects is below. For further inspiration you can also have a look at the list of projects I have supervised in the past, at the bottom of this page.

Example Topics

Estimating the Intensity Function of a Poisson Process

(This is a 30 credit project, but could be extended to 40 credits.)

A Poisson Process is a stochastic process which consists of a collection of (random) points in time. An example of a Poisson process could be the points of time where customers arrive in a shop. The concept of a Poisson process can be generalised to processes with points in arbitrary sets (instead of points in time). Different time interval ls may have different expected numbers of points; this is described by the intensity function $\lambda$: the expected number of points in the time interval $[a,b]$ is given by $$ E\bigl( N[a,b] \bigr) = \int_a^b \lambda(t) \,dt. $$

The task of this project is to study and compare different methods to estimate the intensity function $\lambda$ for a given instance of a Poisson process. A possible extension would be to consider methods for statistically testing the hypothesis that a sample of a Poisson process corresponds to a given intensity function. The project combines theoretical and computational aspects.

Selected references:

• J. F. C. Kingman: Poisson Processes, Clarendon Press, Oxford, 1993.
• B. W. Silverman, Density Estimation, Chapman and Hall, 1986.
• Y. A. Kutoyants, Statistical Inference for Spatial Poisson Processes, Springer, 1998.

Multilevel Monte Carlo Path Simulation

(This project can be taken either as a 40 or a 30 credits project)

Monte Carlo Methods allow to estimate an expectation $E(X)$ numerically by making use of the law of large numbers: $$ E(X) \approx \frac1n \sum_{i=1}^n X_i $$ where the $X_i$ are independent copies of $X$. The result is only approximate, the error decays as $n\to\infty$ and it transpires that the computational cost of approximating the expectation up to a precision of $\varepsilon$ grows like $1/\varepsilon^2$ as $\varepsilon\to 0$. In practical applications, sometimes $X$ and the $X_i$ themselves can be constructed only approximately. In these cases, an additional error is introduced by using approximations $X^{(m)}_i$ instead of $X_i$, where increasing $m$ reduces this error, but also causes additional computational cost.

In the presence of the two kinds of error discussed above, multilevel Monte Carlo methods minimise the computational cost for estimating $E(X)$ by balancing $m$ and $n$ in a clever way. It turns out that the optimal strategy involves generating many samples for small $m$ (where the construction is cheaper) and only a few samples for big $m$ (where the results are more accurate).

The objective of this project is to understand and summarise the underlying theory and to implement the resulting methods. The main focus of this project is theoretical, but obviously programming is also required.

Selected references:

• Michael B. Giles: Multilevel Monte Carlo path simulation, Operations Research, 56(3), pages 607-617, 2008.
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Random Number Generation

(This is a 15 credit project but can be extended to any of the available project types)

Many algorithms in statistics, for example Monte Carlo methods, make use of random numbers. Since computer programs are inherently deterministic (two independent runs of the same program will result in the same output), generating random numbers with a computer is a non-trivial task. This problem is normally solved using pseudo random number generators (PRNGs).

Modern PRNGs consist of two components:

1. one algorithm to create (pseudo) random numbers uniformly on the interval $[0,1]$, and
2. a family of algorithms to convert these random number to a given distribution (like normal, binomial, Poisson, etc.)

The aim of this project is to study methods for use in the second of these steps.

The projects has a theoretical component (understanding why/how the methods work) and a computational component (implementing the methods and determining efficiency). For this topic, the MATH5835 (statistical computing) module is desirable, but not required.

Selected references:

• Donald E. Knuth: Seminumerical Algorithms. Vol. 2 of The Art of Computer Programming. Addison-Wesley, Second edition, 1981.
• G. Marsaglia and W.W. Tsang: The Ziggurat Method for Generating Random Variables. Journal of Statistical Software, vol. 5, no. 8, pp. 1–7, 2000.

Introduction to the Theory of Large Deviations

(This project could be either a 15 credit project or could be extended to be a 30 credit project)

Large deviations theory is the topic of computing the asymptotic behaviour of small probabilities. This provides a refinement of results obtained by the law of large numbers: For example, where the law of large numbers tells us that the average of a series of throws of regular dice converges to 3.5, large deviation theory allows us to to study how fast the probability of staying away from this value goes to 0 as the number of throws increases.

The aim of this project is to understand and summarise the basics of large deviation theory. This is a theoretical project without a programming component, so solid understanding of basic probability will be helpful.

Selected references:

• Amir Dembo and Ofer Zeitouni: Large Deviations Techniques and Applications. Second edition, Springer, vol. 38 of Applications of Mathematics, 1998.
• Frank den Hollander: Large Deviations. American Mathematical Society, vol. 14 of Fields Institute Monographs, 2000.

Past Projects

The following list contains projects which I have supervised in the past or which I am currently supervising.