Georgia Institute of Technology College of Sciences

Many derivatives products are directly or indirectly associated with integrated diffusion processes. We develop a general perturbation method to price those derivatives. We show that for any positive diffusion process, the hitting time of its integrated process is approximately normally distributed when the diffusion coefficient is small. This result of approximate normality enables us to reduce many derivative pricing problems to simple expectations. We illustrate the generality and accuracy of this probabilistic approach with several examples, with emphasis on timer options. Major advantages of the proposed technique include extremely fast computational speed, ease of implementation, and analytic tractability.

A presentation by Wing Li (Math, former NSF program officer) and Marion Usselman (Associate Director of CEISMC).

I mainly discuss the following problem: given a set of scattered locations and nonnegative values, how can one construct a smooth interpolatory or fitting surface of the given data? This problem arises from the visualization of scattered data and the design of surfaces with shape control. I shall start explaining scattered data interpolation/fitting based on bivariate spline functions over triangulation without nonnegativity constraint. Then I will explain the difficulty of the problem of finding nonnegativity perserving interpolation and fitting surfaces and recast the problem into a minimization problem with the constraint. I shall use the Uzawa algorithm to solve the constrained minimization problem. The convergence of the algorithm in the bivariate spline setting will be shown. Several numerical examples will be demonstrated and finally a real life example for fitting oxygen anomalies over the Gulf of Mexico will be explained.

The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. In this talk, I will explain the ideas of the proof and discuss our recent simplifications. One of the main ingredients in the proof is a relative Szemerédi theorem, which says that every relatively dense subset of a pseudorandom set of integers contains long arithmetic progressions. Our main advance is both a simplification and a strengthening of the relative Szemerédi theorem, showing that a much weaker pseudorandomness condition suffices. I will explain the transference principle strategy used in the proof. Also see our recent exposition of the Green-Tao theorem: http://arxiv.org/abs/1403.2957 Based on joint work with David Conlon and Jacob Fox.