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.