# Stochastic Processes

**Characterisation of phase-type distributions**

(posted for 2017/2018 - Also posted under "Operations Research")

A random variable that is defined as the absorption time of a finite-state continuous-time Markov chain with a single absorbing state is said to have a phase-type distribution. The representation of a phase-type distribution is the pair (a,T) where a is the initial state probability vector and T is the infinitesimal generator for the n transient states of the Markov chain. The order of the representation is n. If the representation is of minimal order, then n is known as the order of the distribution.

The Laplace-Stieltjes transform of a phase-type distribution is a rational function. If the numerator and denominator polynomials have no factors in common, the degree of the dominator is called the algebraic degree of the phase-type distribution. In general, the algebraic degree of a phase-type distribution is no larger than its order.

There are a number of open problems concerning the connection between the algebraic degree and the order of phase-type distributions. For example, it is well known that there exist phase-type distributions that have small algebraic degree but very large order. The problem is, however, given a phase-type distribution with a certain algebraic degree, how can we determine its order? And if we can determine its order, how can we find a minimal representation for it?

In this project we will investigate some of these open problems. This project is theoretical in nature and fits in with both Operations Research and Stochastic processes. Some familiarity with coding, particularly in Matlab, will come in handy, but is not essential.

__Contact: Mark Fackrell fackrell@unimelb.edu.au

**Stochastic models for populations with a carrying capacity**

(posted for 2017/2018)

Many biological populations experience a logistic growth, that is, the population per capita growth rate decreases as the population size approaches a maximum imposed by limited resources in the environment, known as the carrying capacity.

Picture from http://bio1510.biology.gatech.edu/module-2-ecology/population-ecology/

A particularly interesting real-world example is the Chatham Island black robin population, an endangered bird species which was saved from the brink of extinction in the 1980's, and whose total size is currently lingering around 250 individuals. The exact reason of the black robin population growth decrease, and the value of the carrying capacity are not yet well understood.

Source: Euan Kennedy's PhD thesis

The main objectives of this project are to study different stochastic models of population-size dependent branching processes, and to develop parameter estimation methods to fit these models to the black robins data. In particular, this will involve

- comparing discrete-time and continuous-time models,
- studying different model outputs such as the distribution of the time until extinction and the total progeny size,
- investigating the sensitivity of the model outputs with respect to the choice of the offspring distribution.

These objectives will be tackled using a combination of simulation studies and theoretical developments.

**Contact**Sophie Hautphenne sophiemh@unimelb.edu.au

For more information on this research group see:

Stochastic Processes

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