An Information Geometry Problem in Mathematical Finance

Abstract

Familiar approaches to risk and preferences involve minimizing the expectation $E_{{\mathrm{I}\!\mathrm{P}}}(X)$ of a payoff function X over a family $\varGamma$ of plausible risk factor distributions ${\mathrm{I}\!\mathrm{P}}$. We consider $\varGamma$ determined by a bound on a convex integral functional of the density of ${\mathrm{I}\!\mathrm{P}}$, thus $\varGamma$ may be an I-divergence (relative entropy) ball or some other f-divergence ball or Bregman distance ball around a default distribution ${{\mathrm{I}\!\mathrm{P}}_0}$. Using a Pythagorean identity we show that whether or not a worst case distribution exists (minimizing $E_{\mathrm{I}\!\mathrm{P}}(X)$ subject to ${\mathrm{I}\!\mathrm{P}}\in \varGamma$), the almost worst case distributions cluster around an explicitly specified, perhaps incomplete distribution. When $\varGamma$ is an f-divergence ball, a worst case distribution either exists for any radius, or it does/does not exist for radius less/larger than a critical value. It remains open how far the latter result extends beyond f-divergence balls.

ICs acknowledges support by the Hungarian National Foundation for Scientific Research OTKA, grant No. K105840. TB acknowledges support by the Josef Ressel Centre for Applied Scientific Computing.