Applied Maths Honours Seminar

Applied Maths Honours Seminar 2010 

Friday, September 17, AGR Carslaw 829: 

3:00 John Mitry 
3:30 Hoan Xuan Nguyen 

4:00 Break 

4:15 John Maclean 
4:45 David Lewis 

***** 
Speaker 1: John Mitry 
Title: Models of HIV infection 

Abstract: Human infection by HIV (Human Immunodeficiency Virus) is a significant
contributor to mortality across the world, particularly throughout South Africa and
Asia.  While at first the infection has a clinical presentation like most other viral
infections, the body is usually unable to completely eliminate the virus, and after 6 -
15 years without symptoms the infection develops into AIDS (Acquired Immunodeficiency
Syndrome).  Despite a detailed clinical knowledge, we still have a limited understanding
of the mechanism by which HIV infection leads to AIDS.  This talk will focus on one
proposed mathematical model which attempts to capture the dynamics of HIV infection.
Following an analysis of the model, the implications and shortcomings, both mathematical
and physiological, are presented.  

***** 
Speaker 2: Hoan Xuan Nguyen 
Title: Inflation-indexed derivatives: modelling and pricing 

Abstract: The purpose of this thesis is to study and discuss some of the most popular
models in pricing inflation-indexed derivatives.  We present the two main approaches in
the literature: the foreign-currency analogy and the market models approach.  The
pricing model introduced by Jarrow and Yildirim which uses the Heath-Jarrow-Morton
framework is reviewed.  The methodology is applied to derive the closed-form solution to
the price of a European call option on the inflation-index and also to price some
inflation-indexed swaps.  Along the second approach, we discuss the lognormal LIBOR
market model.  Subsequently, in this setting, we price Year-on-Year Inflation-indexed
Swaps and inflation caplets and floorlets.  

***** 
Speaker 3: John Maclean 
Title: Optimal timing and the role of uncertainty 

Abstract: Two papers on the topic of pollution reduction give conflicting results
regarding the effect of uncertainty on the timing of a reduction in emissions.  Each
paper produces a model with different underlying processes and a different focus.  Each
model is solved by dynamic programming to produce two regions obeying certain
differential equations; the solution involves the consideration of a free boundary
problem.  The more surprising set of results is extended in some cases, and commentary
will follow; a mistake in one of the papers is partially responsible for the different
predictions they make.  

***** 
Speaker 4: David Lewis 
Title: Stochastic homogenisation on manifolds 

Abstract: We study a slow-fast timescale separated toy model of a charged particle with
a small mass in a magnetic field.  Our goal is to extract a reduced equation for the
effective slow dynamics.  This talk represents the first step towards achieving this
goal.  We take a stochastically perturbed Hamiltonian system in which noise is used to
model chaotic dynamics.  We ensure energy is conserved by projecting the stochastic
noise onto the energy manifold.  Stochastic singular perturbation theory is applied to
this system to derive an evolution equation for the lowest order behaviour.  We thus
extract a reduced effective Stochastic Differential Equation.