By Prof. Ioannis KARATZAS
Department of Mathematics and Department of Statistics, Columbia University, New York
Venue : Natural Science Bldg Auditorium (E6-1 1501), KAIST
May 19 (TUE), 4:00 pm
CONSERVATIVE DIFFUSIONS AS ENTROPIC FLOWS OF STEEPEST DESCENT
Abstract: We provide a detailed, probabilistic interpretation for the variational characterization of conservative diffusions as entropic flows of steepest descent. Jordan, Kinderlehrer, and Otto showed in 1998, via a numerical scheme, that for diffusions of Langevin-Smoluchowski type the Fokker-Planck probability density flow minimizes the rate of relative entropy dissipation, as measured by the distance traveled in terms of the quadratic Wasserstein metric in the ambient space of configurations. Using a very direct perturbation analysis we obtain novel, stochastic-process versions of such features; these are valid along almost every trajectory of the motion in both the forward and, most transparently, the backward, directions of time. The original results follow then simply by “aggregating”, i.e., taking expectations. As a bonus we obtain a version of the HWI inequality of Otto and Villani, relating relative entropy, Fisher information, and Wasserstein distance.
Joint work with W. Schachermayer, B. Tschiderer and J. Maas (Vienna); we report also on related work of L.Yeung and D. Kim.
May 22 (FRI), 4:00 pm
PORTFOLIO THEORY AND ARBITRAGE
Abstract: We develop a mathematical theory for finance based on the following “viability” principle: That it should not be possible to fund a non-trivial liability starting with arbitrarily small initial capital. In the context of continuous asset prices modeled by semimartingales, we show that proscribing such egregious forms of what is commonly called “arbitrage” (but allowing for the possibility that one portfolio might outperform another), turns out to be equivalent to any one of the following conditions:
(i) a portfolio with the local martingale numeraire property exists,
(ii) a growth-optimal portfolio exists,
(iii) a portfolio with the log-optimality property exists,
(iv) a local martingale deflator exists,
(v) the market has locally finite maximal growth.
We assign precise meaning to these terms, and show that the above equivalent conditions can be formulated entirely, in fact very simply, in terms of the local characteristics (the drifts and covariations) of the underlying asset prices. Full-fledged theories for hedging and for portfolio/consumption optimization can then be developed, as can the important notion of “market completeness”.
Book with the same title with C. Kardaras (London).
May 28 (THU), 4:00 pm
THE HEREDITARY LAWS OF LARGE NUMBERS
Abstract: The celebrated theorem of Komlos (1967) establishes L^1-boundedness as a sufficient condition for a sequence of measurable functions on a probability space to contain a subsequence along which, and along whose every further subsequence (“hereditarily”), the Cesaro averages converge to a “randomized mean” in the spirit of the Strong law of Large Numbers. We provide conditions not only sufficient, but also necessary, for this result, as well as for the hereditary analogues of the Weak Law of Large Numbers, of the Hsu-Robbins-Erdos Law of Large Numbers, and of the Law of the Iterated Logarithm.
Joint work with I. Berkes (Budapest) and W. Schachermayer (Vienna).