Ricardas Zitikis (University of Western Ontario, Canada)
X
“The dose makes the poison,” said Paracelsus, the father of toxicology. Thinking of risk, how much of it is good or bad? Despite all the illuminating and far-reaching results available in the literature on the topic, I will go back to the very basics: first, I will briefly touch upon risk preferences, then look at some elusive behavior of larger risks, such as insurance losses above deductibles, and finally venture into extremes. Though somewhat philosophical in nature, the talk is based on very recent results published by the speaker and his co-authors in journals such as the Annals of Actuarial Science (2015, 9(1), 58-71), Insurance: Mathematics and Economics (2016, 69(July), 97-10), and ASTIN Bulletin - The Journal of the International Actuarial Association (2015, 45(3), 661-678).
Measuring risks – a tricky task with a lot of scientific fun
Oct. 27, 2016 2pm (Math 350)
Functional Analysis
Arlan Ramsay (CU Boulder)
X
Robert Richtmyer, a former member of this department, thought that it would be good to teach axiomatic Euclidean and hyperbolic geometry using axioms that treated the real numbers as given and used some of their properties as tools for proving the theorems. He made notes for his classes, and I assisted him in turning the notes into a book.
I want to outline some of the basic results and discuss how facts about continuity from undergraduate real analysis can be used to prove existence and uniqueness of Euclidean and hyperbolic planes. This is in the spirit of differential geometry, but with minimal use of derivatives. We do manage to avoid some of the very clever purely geometric arguments found in most other treatments of these topics.
Using Continuity Arguments in Plane Geometry
Oct. 27, 2016 3pm (MATH 350)
Probability
Ricardas Zitikis (University of Western Ontario, Canada)
X
The notion of monotone rearrangement of functions has been known for a long time and can be traced back to the classical book on “Inequalities” by G. H. Hardy, J. E. Littlewood, and G. Pólya. The notion of convex rearrangement of functions originates from a work of Yu. Davydov published several decades ago. To illustrate, the quantile function in Statistics is a monotone rearrangement, whereas the Lorenz curve in Econometrics is a convex rearrangement. In this talk I will discuss an index of monotonicity that has naturally arisen from these rearrangements and provided a useful tool for comparing functions, stochastic processes, and other objects according to their monotonicity properties. I will hint at several problems in Insurance and Finance, and in particular in Education, where such comparisons are of interest.
Monotone and convex rearrangements: mathematics, probability, and applications