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Research Theme: Risk Modelling


Theme Leader: Kostya Borovkov (UM)
CIs: Richard Brak (UM), Tony Guttmann (UM), Peter Hall (UM), Aleks Owczarek (UM), Phil Pollett (UQ), Peter Taylor (UM), Honorary Professorial Fellow: Ian Enting (UM).

All complex systems are subject to operational risks, especially those associated with extreme behaviour. The analysis and prediction of extreme behaviour emerges from the influence of small scale phenomena on large scale behaviour.

Goal: The fast and efficient quantification of risk.

Illustrative problem: The effective pricing of financial and other derivatives, and of carbon credits.

Applications: Insurance, power networks and pest control.

The current list of major projects was confirmed at the recent Theme Workshop, held in connection with the Annual Meeting of the Australian Mathematical Society, La Trobe University, Monday 24th September 2007. They are as follows:

• Species dispersal: extension of standard diffusion models to more realistic ones, allowing jumps;
• Evaluating expected time to `functional extinction' of a biological population;
• Deterministic and probabilistic methods for valuation of exotic options;
• Interaction of financial risk and insurance risk;
• Asset price modeling and applications;
• Flow through porous media;
• Transport.

Introduction

Any complex system, be it a computer program, the Internet, a financial system or a high-rise building is subject to various kinds of risks associated with different aspects of its functioning. Taking into account the fact that most such systems operate under conditions of uncertainty, the most appropriate models for studying such risks are stochastic models in which a swarm of random events represent undesirable outcomes of different severity. In such a formulation, undesirable events are usually identified with the process either entering certain regions of the state space, experiencing large changes in its value or passing through a critical point (see Research Theme 1).

Most of the risk evaluation procedures that are currently used are based on elementary, and therefore crude and often inappropriate, stochastic models, for which the analytic solutions of the relevant problems are known. An example is the currently-accepted model for finding the probability of ruin of an insurance company. Alternatively, simulation is used to estimate the quantities of interest. Both approaches give unreliable answers and there is substantial pressure, from both applied and theoretical viewpoints, to extend the class of stochastic models for which either analytical or efficient numerical solutions are available.

The Centre aims to concentrate its efforts on attacking a number of important problems in risk analysis. They include boundary crossing problems for stochastic process models in insurance and reliability contexts. Of particular significance are models taking into account random interest rate environments. Further problems of interest occur in the context of modelling and analysing risks of loan-default or company collapse, which, in view of the recent events in the corporate world, have obvious significance. Projects undertaken by the Centre's researchers in this areas include:

Risk Modelling — 2004

ROC CURVE ESTIMATION
Peter Hall (Chief Investigator), Rob Hyndman (Monash University)

A Receiver Operating Characteristic (ROC) curve is used in tests for the presence of particular conditions such as diseases. They are very common in the field of medical statistics. Essentially a ROC curve is a plot of the true positive rate against the false positive rate as different threshold values of a diagnostic test are used. Hall and Hyndman have proposed a method for bandwidth selection when estimating ROC curves.

Risk Modelling — 2004

ILL-POSED PROBLEMS
Peter Hall (Chief Investigator), R. Paige, F.H. Ruymgaart, J.Y. Yin

This project involves work on methods for solving two classes of ill-posed problems. The researchers have developed a wavelet method for solving Abel-type equations and non-parametric methods for deconvolving multiperiodic functions.


Risk Modelling — 2006- ongoing

CLIMATE CHANGE
Ian Enting (Professorial Fellow) and Nathan Clisby (RF)


Analysis of projections of climate change.

(2007) Climate benefits of geosequestration of CO2, i.e. the capture, extraction, separation, collection, etc, of carbon dioxide and a means for its storage or use, have been analysed, taking into account a range of uncertainty in efficiency of capture and potential leakage. These results are being used by CSIRO in designing strategies to monitor a pilot geosequestration project in the Otway Basin.

(2007-8). Studies of Emission trajectories leading to stabilisation of greenhouse gas concentrations as input to the Garnaut Review of Climate Change. This is being undertaken with CASPI (Climate Adaptation - Science and Policy Initiative), University of Melbourne.


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