Information geometry in mathematical finance: Model risk, worst and almost worst scenarios

The mathematical problem addressed is minimising the expectation of a random variable over a set of feasible distributions P ϵ Γ, given as a level set of a convex integral functional. As special cases, Γ may be an f-divergence or f-divergence ball or a Bregman ball around a default distribution. Our approach is motivated by geometric intuition and relies upon the theory of minimising convex integral functionals subject to moment constraints. One main result is that all “almost minimisers” P ϵ Γ belong to a small Bregman ball around a specified distribution or defective distribution P, equal to the strict minimiser if that exists but well defined also otherwise.