Mathematical Sciences at the University of Liverpool

Sample PhD Projects in Stochastics

General Interests

Markov processes; stochastic analysis; stochastic epidemic models; optimal control theory; Bayesian statistics; non-parametric statistics


Dr Gabriel Berzunza Ojeda

I am interested in branching structures and their applications:

Random tree structures - The purpose of this project is to develop new probabilistic techniques to provide a thorough description of various aspects of (random) trees in different random settings. For example, describing the structure of typically large (random) trees, and more generally large graphs. Beyond the purely probabilistic or combinatorial aspects, the goal also is to study several other models that come from:

1. Biology to understand the spread of epidemics

2. Theoretical computer science in the analysis of algorithms and study of data structures

3. Statistical physics such as percolation

4. Probabilistic aspects of evolutionary biology - This project is devoted to the study of interactions between organisms and their environment which influence their reproductive success and contribute to genotype and phenotype variation which is one of the main questions in evolutionary ecology and population genetics. The understanding of such evolutionary-ecological processes is very complicated and it requires the substantial use of mathematical models and methods. These include branching processes, superprocesses, fragmentation-coalescence processes, Lévy processes, stochastic partial differential equations, etc. Typical questions related to this subject of research might concern the growth rate of interacting populations, the probability a certain sub-population survives or perishes, genealogies of certain individuals in some population, the effects of mutation and recombination of genetic material, reproduction mechanisms, competition, cooperation and selection.


 Dr Kai Liu

1. Ruin probabilities for Lévy processes

This project is interested in using Lévy process for risk reserve modelling.

2. Optimal Consumption under partial Observations for a Stochastic Systems with Delay

In this project, we introduce time delay into the stochastic system which allows us to take into account the fact that it may take some time before new market information affects the value of our investment. Also the decisions we make regarding consumption may be based on both present and past values of the wealth. The second case captures in addition the fact that we do not always have complete information about all the parameters in a mathematical model in finance.

3. Robust techniques for pricing and hedging options

The classical approach to the pricing and hedging of financial options - pioneered by Black and Scholes, is to postulate a model for the underlying asset, and use the notion of risk-neutral pricing to find the arbitrage-free price of derivative contracts based on the underlying. The recent financial crisis has demonstrated the frailty of such model based techniques, and highlighted the need for more robust methods of hedging and pricing derivative contracts. A starting point for an alternative approach is to ask: given a set quoted prices, when are these prices consistent with some model? This proves to be the starting point for a number of interesting questions, and a project in this area would look to investigate some of these.

4. Stability of infinite dimensional stochastic systems

This project is devoted to the investigation of various criteria, theoretical or practical, of stochastic stability and their possible applications in popular models like stochastic reaction-diffusion equations and stochastic volatility in option pricing with very large number of incremental market activities.

5. Large deviation of stochastic systems with memory

Over the last three decades, large deviation problem for stochastic evolution equations, especially stochastic partial differential equations, has been extensively investigated by many researchers. In this project, we will deal with the large deviation principle for families of probability measures associated with the stochastic retarded functional evolution equations.


Dr Alexey Piunovskiy

Optimal Control: theory and applications.

Many real life phenomena are described by ordinary differential equations. Others can be approximated by Markov chains or more complicated random processes. If we can affect dynamically this or that parameter then we deal with a controlled dynamical system. Generally speaking, the problem is to find the best, optimal control. Remember, many problems are multiple objective.
The aim of such a project for students can be: to justify the problem statement, to study optimisation methods like dynamic programming and Pontryagin's maximum principle, to investigate analytically (or numerically) a particular version of the problem formulated, to undertake computer simulations, to compare different mathematical models describing the same real life phenomenon.

Projects:

1. Convex Analytic Approach to certain controlled Markov chains and jump processes

2. Analysis of communication networks

3. Approximations of controlled birth-and-death processes


Dr Linglong Yuan

I’m interested in, generally speaking,  the probability theory and its applications.

Currently I’m focusing on the following 4 themes:

1. Coalescent theory, branching processes, random trees, measure valued processes;

2. Stochastic modelling in biology​, Data-driven computations;

3. Condensation in stochastic models;

4. Exchangeability, Extendibility of finite exchangeable sequences

Possible PhD projects are:

1. Finer properties of coalescent processes and branching processes;

2. Stochastic modelling and statistical inference for biological experiments;

3. Stochastic modelling of diploid populations;

4. Exchangeable stochastic networks and its applications;

5. Condensation phenomena in complex stochastic networks.