TITLE: Knightian Uncertainty and Nonlinear Expectations

SPEAKER: Professor Shige Peng, Shandong University

ABSTRACT:

A. N. Kolmogorovs Foundations of the Theory of Probability published in 1933, has established the modern axiomatic foundations of probability theory. Since then this theory has been profoundly developed and widely applied to situations where uncertainty cannot be neglected.

But in 1921 Frank Knight has been already clearly classified two types of uncertainties: the first one is for which the probability is known; the second one, now called Knightian uncertainty, is for cases where the probability itself is also uncertain. The situation with Knightian uncertainty has become one of main concerns in the domain of data processing, economics, statistics, and specially in measuring and controlling financial risks. A long time challenging problem is how to establish a theoretical framework, comparable to the Kolmogorovs one, to treat these more general situations with Knightian uncertainties. Motivated from the measure of risk, the objective of the theory of nonlinear expectation rapidly developed in recent years is to solve this problem.

This is a huge program. Some fundamental results have been established and well-understood such as law of large numbers, central limit theorem, martingales, G-Brownian motions, G-martingales and the corresponding stochastic calculus of It?os type, nonlinear Markov processes, as well as the calculation of measures of risk in finance. But many intersting problems are still to be explored, e.g., statistics in the framework nonlinear expectations.

Bio:

Shige Peng is one of the most original and active contributors to the field of probability theory and financial mathematics. Recognized around the world as a leading figure for his work on backward stochastic differential equations and nonlinear expectations, he has had a profound impact in both mathematics and financial engineering. A member of the Chinese Academy of Science, he has been a plenary speaker at the International Congress of Mathematicians, one of the highest honors given to mathematicians.

As a prominent figure in the Chinese mathematics and financial engineering communities he has held numerous visiting professorships and lectured at major institutions and research conferences worldwide, including the Ecole Polytechnique, as well as Osaka, Tokyo, Columbia, Brown, and Princeton universities. He is a Professor First Class at the Institute of Mathematics, Shandong University, and a Distinguished Professor of Ministry of Education of China.

In the course of his visits to Princeton as a Global Scholar in the departments of Mathematics, Operations Research and Financial Engineering, and the Program in Applied and Computational Mathematics, Peng will help nucleate collaborative activities and research interests in the area of stochastic analysis and applications to financial mathematics. He will teach short courses on backward stochastic differential equations and the theory of nonlinear expectations, help organize formal and informal seminars on probability theory and financial mathematics, co- advise undergraduate independent work and graduate students in Mathematics, ORFE and PACM, and collaborate on research with colleagues on campus.