Abstract:
We study models for large equity markets, whose stability properties are in broad agreement with observations on the distribution of capital in the U.S. equity markets over the period 1920-2010.
These models assign, to the different stocks, growth rates and variances that depend both on name and on relative capitalization rank. The models are able realistically to capture the observed stability of capital distribution, while being simple enough to allow for rather detailed analytic study. The methodologies used in this study touch upon the question of triple points for systems of interacting diffusions; in particular, some choices of parameters may permit triple (or higher-order) points to occur, but we show that this has no resultant effect on any of the stability properties. The theory we develop has connections with the analysis of Queueing Networks in heavy traffic, as well as with models of competing partile systems in Statistical Mechanics and with the Sherrington-Kirkpatrick model for spin-glasses.
REFERENCES
[1] T. Ichiba and I. Karatzas. On collisions of Brownian particles. Annals of
Applied Probability 20 (2010), 951-977.
[2] T. Ichiba, V. Papathanakos, A. Banner, I. Karatzas and E.R. Fernholz. Hybrid
Atlas Models. Annals of Applied Probability, to appear (2011).
Bio:
Ioannis Karatzas studied at the Ionideion Gymnaseion in Piraeus (Bacc. 1970), at the
National Technical University of Athens (Diploma 1975), and at Columbia University
(M.Sc. 1976; M.Phil. 1978; Ph.D. 1980). He is the Eugene Higgins Professor of Applied
Probability at Columbia's Department of Mathematics.
He has held visiting positions at Brown (1979-80), MIT (1984-85), Rutgers (1989), the Courant Institute of NYU (1994-95), Yale (2001-02), and at the Universities of North Carolina (spring 1986), Paris (fall 1986) and Pennsylvania (fall 1992). He held the Andre Aisenstadt Chair at the Centre de Recherches Mathematiques, University of Montreal in spring of 1996. He works on Probability Theory and Mathematical Statistics, Random Processes, Stochastic Control, Mathematical Economics and Finance. He is a Fellow of the Institute of Mathematical Statistics and has served as managing editor for the Springer-Verlag book series "Applications of Mathematics", and on the editorial boards of journals such as "Applied Mathematics & Optimization", "Stochastics", "The Annals of Applied Probability", "Electronic Journal of Probability", "Mathematical Finance", the "SIAM Journal on Mathematical Analysis" and the "SIAM Journal on Control & Optimization".
He has written (with Steven E. Shreve) the book "Brownian Motion and Stochastic Calculus". First published by Springer-Verlag in 1987 this book, now in its tenth printing, is the standard reference in the field of Stochastic Analysis. His monograph "Methods of Mathematics of Finance" (Springer-Verlag, August 1998, again with Shreve) is the authoritative treatise of a burgeoning new field. His 18 former Ph.D. students are now on the faculties of Boston University, Brown, Caltech, Columbia, CUNY, NYU, Princeton; the Universities of Texas, Alberta, Kyoto, Reykjavik and Washington; or in various industrial positions.
He has been invited to give an IMS/Bernoulli Society Lecture, the Andre Aisenstadt Lectures at the University of Montreal (published by the American Mathematical Society in 1996), the Lezioni Calileiane at the Scuola Normale Superiore in Pisa, the Matsushita Lecture at Fudan University in Shanghai, the Kuwait Lecture at Cambridge University, the Leaders in Mathematical Sciences Lecture at IBM Yorktown Heights, the Wolfe Lecture at Rice University, the Ziwet Lectures and the inaugural Michael Woodroofe Lecture at the University of Michigan. He has been Plenary Speaker at numerous Conferences, including the World Congress of the Bachelier Society, the HEC/Montreal Optimization Days, the "Stochastic Processes and Applications" Conference, the Midwest Probability Colloquium, the International Symposium on "Stochastic Analysis and Related Topics" in honor of Prof. Kiyoshi Ito's 88th Birthday at Kyoto University, September 2002, and the 2005 Abel Symposium in Oslo, Norway. In May 2005 he was awarded a Doctorate Honoris Causa by the Athens University Faculty for Economics and Commerce.