Emanuela Rosazza Gianin

Putting order in risk measures

Abstract This paper introduces a set of axioms that define convex risk measures. Duality theory provides the representation theorem for these measures and the link with pricing rules.

Law invariant convex risk measures

As a generalization of a result by Kusuoka (2001), we provide the representation of law invariant convex risk measures. Very particular cases of law invariant coherent and convex risk measures are also studied.

Representation of the penalty term of dynamic concave utilities

In the context of a Brownian filtration and with a fixed finite time horizon, we provide a representation of the penalty term of general dynamic concave utilities (hence of dynamic convex risk measures) by applying the theory of g-expectations.

Dual Representation of Minimal Supersolutions of Convex BSDEs

We give a dual representation of minimal supersolutions of BSDEs with non-bounded, but integrable terminal conditions and under weak requirements on the generator which is allowed to depend on the value process of the equation. Conversely, we show that any dynamic risk measure satisfying such a dual representation stems from a BSDE. We also give a condition under which a supersolution of a BSDE is even a solution.

ON THE PENALTY FUNCTION AND ON CONTINUITY PROPERTIES OF RISK MEASURES

We discuss two issues about risk measures: we first point out an alternative interpretation of the penalty function in the dual representation of a risk measure; then we analyze the continuity properties of comonotone convex risk measures. In particular, due to the loss of convexity, local and global continuity are no more equivalent and many implications true for convex risk measures do not hold any more.

Capital allocation à la Aumann–Shapley for non-differentiable risk measures

We study capital allocation rules satisfying suitable properties for convex and quasi-convex risk measures, by focusing in particular on a family of capital allocation rules based on the dual representation for risk measures and inspired by the Aumann–Shapley allocation principle. These rules extend some well known methods of capital allocation for coherent and convex risk measures to the case of non-Gateaux-differentiable risk measures. We also analyze the properties of the allocation principles here introduced and discuss their suitability in the quasi-convex context.