On the representation of weakly maxitive monetary risk measures and their rate functions

TL;DR

This paper generalizes the Gärtner-Ellis large deviations theorem by representing weakly maxitive monetary risk measures through rate functions on arbitrary sets, extending large deviation principles to sublinear expectations.

Contribution

It introduces a novel representation for weakly maxitive monetary risk measures, broadening the scope of large deviations theory beyond classical dual spaces.

Findings

Provides a new functional analytic representation of risk measures.

Establishes a large deviation principle for sublinear expectations.

Extends the Gärtner-Ellis theorem to more general settings.

Abstract

The present paper provides a representation result for monetary risk measures (i.e., monotone translation invariant functionals) satisfying a weak maxitivity property. This result can be understood as a functional analytic generalization of G\"{a}rtner-Ellis large deviations theorem. In contrast to the classical G\"{a}rtner-Ellis theorem, the rate function is computed on an arbitrary set of continuous real-valued functions rather than the dual space. As an application of the main result, we establish a large deviation result for sequences of sublinear expectations on regular Hausdorff topological spaces.