An Information Geometry Problem in Mathematical Finance

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.