Date: Thursday, October 25, 2012
Location: 4096 East Hall (3:00 PM to 4:00 PM)
Title: Can one construct nonlinear conditional expectations?
Abstract: The theory of nonlinear expectations, particularly the G-expectations of S. Peng, has been popular in recent years. Partially inspired by risk measures and by stochastic control problems, a nonlinear expectation can be viewed as the worst-case expectation of a random variable under model uncertainty. One of the most basic problems that arise in this setting is the definition of nonlinear conditional expectations. Even in the simplest interesting case---martingales with volatility uncertainty---previous constructions can only define conditional expectations for (quasi)continuous random variables, which is hardly satisfactory both conceptually and in practice. The aim of this talk is to show how one can construct nonlinear conditional expectations for Borel random variables. While the construction is very simple, such conditional expectations are in general no longer Borel, but only universally measurable. Unfortunately, it is fundamentally impossible to construct conditional nonlinear expectations for universally measurable random variables, as will be illustrated by an unpleasant counterexample. Therefore, "nonlinear probability theory" is destined to remain much more limited than its linear counterpart that we all know and love. (Joint work with M. Nutz)
Speaker: Ramon van Handel
Institution: Princeton University
Event Organizer: Erhan Bayraktar erhan@umich.edu
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