Max-stable random sup-measures with comonotonic tail dependence

Several objects in the Extremes literature are special instances of max-stable random sup-measures. This perspective opens connections to the theory of random sets and the theory of risk measures and makes it possible to extend corresponding notions and results from the literature with streamlined proofs. In particular, it clarifies the role of Choquet random sup-measures and their stochastic dominance property. Key tools are the LePage representation of a max-stable random sup-measure and the dual representation of its tail dependence functional. Properties such as complete randomness, continuity, separability, coupling, continuous choice, invariance and transformations are also analysed.Comment: 28 pages, 1 figur

Random correspndences as bundles of random variables.

We prove results that relate random correspondences with their measurable selections, thus providing a foundation for viewing random correspondences as "bundles" of random variables.

How regular can maxitive measures be?

We examine domain-valued maxitive measures defined on the Borel subsets of a topological space. Several characterizations of regularity of maxitive measures are proved, depending on the structure of the topological space. Since every regular maxitive measure is completely maxitive, this yields sufficient conditions for the existence of a cardinal density. We also show that every outer-continuous maxitive measure can be decomposed as the supremum of a regular maxitive measure and a maxitive measure that vanishes on compact subsets under appropriate conditions.Comment: 24 page

Ultramodular functions.

We study the properties of ultramodular functions, a class of functions that generalizes scalar convexity and that naturally arises in some economic and statistical applications.

On convexity and supermodularity.

Concavity and supermodularity are in general independent properties. A class of functionals defined on a lattice cone of a Riesz space has the Choquet property when it is the case that its members are concave whenever they are supermodular. We show that for some important Riesz spaces both the class of positively homogeneous functionals and the class of translation invariant functionals have the Choquet property. We extend in this way the results of Choquet [1] and Konig [4].

Law-Invariant Functionals that Collapse to the Mean: Beyond Convexity

We establish general "collapse to the mean" principles that provide conditions under which a law-invariant functional reduces to an expectation. In the convex setting, we retrieve and sharpen known results from the literature. However, our results also apply beyond the convex setting. We illustrate this by providing a complete account of the "collapse to the mean" for quasiconvex functionals. In the special cases of consistent risk measures and Choquet integrals, we can even dispense with quasiconvexity. In addition, we relate the "collapse to the mean" to the study of solutions of a broad class of optimisation problems with law-invariant objectives that appear in mathematical finance, insurance, and economics. We show that the corresponding quantile formulations studied in the literature are sometimes illegitimate and require further analysis

Martingale optimal transport duality

We obtain a dual representation of the Kantorovich functional defined for functions on the Skorokhod space using quotient sets. Our representation takes the form of a Choquet capacity generated by martingale measures satisfying additional constraints to ensure compatibility with the quotient sets. These sets contain stochastic integrals defined pathwise and two such definitions starting with simple integrands are given. Another important ingredient of our analysis is a regularized version of Jakubowski's $S$-topology on the Skorokhod space.Comment: 29 page

Law-invariant functionals that collapse to the mean: Beyond convexity

We establish general "collapse to the mean" principles that provide conditions under which a law-invariant functional reduces to an expectation. In the convex setting, we retrieve and sharpen known results from the literature. However, our results also apply beyond the convex setting. We illustrate this by providing a complete account of the "collapse to the mean" for quasiconvex functionals. In the special cases of consistent risk measures and Choquet integrals, we can even dispense with quasiconvexity. In addition, we relate the "collapse to the mean" to the study of solutions of a broad class of optimisation problems with law-invariant objectives that appear in mathematical finance, insurance, and economics. We show that the corresponding quantile formulations studied in the literature are sometimes illegitimate and require further analysis

The convexity-cone approach to comparative risk and downside risk.

We establish a calculus characterization of the core of supermodular games, which reduces the description of the core to the computation of suitable Gateaux derivatives of the Choquet integrals associated with the game. Our result generalizes to infinite games a classic result of Shapley (1971). As a secondary contribution, we provide a fairly complete analysis of the Gateaux and Frechet differentiability of the Choquet integrals of supermodular measure games.