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Exponentially concave functions and a new information geometry

Banff International Research Station for Mathematical Innovation and Discovery

Exponentially concave functions are concave functions whose exponentials are also concave. The gradient map of such functions over the unit simplex appear as functionally generated portfolios introduced by Fernholz. These gradient maps are also solutions of a remarkable Monge-Kantorovich optimal transport problem. Suppose we are trading a functionally generated portfolio in discrete time, in the absence of transaction costs. What is the optimal frequency of trading? Contrary to popular beliefs, more frequent trading is not necessarily better. The answer lies in a new information geometry which is an exponential version of the celebrated classical information geometry of Bregman divergence. This new geometry is not flat, and yet, the geodesics satisfy a Pythagoras theorem. The answer to the optimal frequency of trading lies in studying this Pythagoras theorem. (Based on joint work with Leonard Wong.)

Mathematics; Game theory, economics, social and behavioral sciences; Probability theory and stochastic processes

video/mp4

Author affiliation: University of Washington

2017-01-26

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/59770

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/