Random concave functions

Abstract:

Concave and convex functions arise naturally in various applications. In this talk we are 

interested in constructing random realizations of these objects, i.e., probability measures 

on spaces of concave functions. First we motivate this problem with nonparametric 

Bayesian statistics and T. Cover's universal portfolio in connection with stochastic 

portfolio theory. In the second part of the talk we study a model of generating random concave

 functions on the simplex by taking the minimum of independent random hyperplanes. We present 

interesting results about the limiting distribution as the number of hyperplanes tends to infinity, 

and show that it can be described via duality by Poisson point processes.

This is on-going work with Peter Baxendale (USC), Christa Cuchiero (Vienna) and Walter Schachermayer (Vienna).