arXiv subjects

Mücahit Aygün

Publications and source records attributed to Mücahit Aygün.

5 recordsLinked to original sources

arXiv2026

Generalized Orlicz premia

We introduce a generalized class of Orlicz premia based on possibly nonconvex loss functions, extending the classical framework of Haezendonck and Goovaerts (1982). Without the usual convexity requirement on the loss function $Φ$, the Orlicz framework naturally encompasses quantiles, expectiles and $L^p$-quantiles while preserving the fundamental properties of Orlicz premia. We show that within this framework cash-additivity axiomatizes $L^p$-quantiles, generalizing the classical `collapse-to-the-mean' result for cash-additive convex Orlicz premia into a `collapse-to-$L^p$-quantiles' result, with expectiles as a special case. We focus on two natural classes of nonconvex loss functions: concave-convex Orlicz functions, which mimic the idea of S-shaped value functions in prospect theory, and GA-convex Orlicz functions, which can be described in terms of comparative convexity with respect to a logarithmic reference, and for which the corresponding Orlicz premium is geometrically convex. Finally, we show that a suitable subclass of generalized Orlicz premia coincides with the class of law-invariant, monotone, positive, positively homogeneous, normalized functionals that are weakly lower semicontinuous, continuous from above, and whose level sets are convex with respect to mixtures (the so-called CxLS property).

q-fin.RM↗

arXiv2025

Dual representations for quasiconvex compositions with applications to systemic risk measures

Motivated by the problem of finding dual representations for quasiconvex systemic risk measures in financial mathematics, we study quasiconvex compositions in an abstract infinite-dimensional setting. We calculate an explicit formula for the penalty function of the composition in terms of the penalty functions of the ingredient functions. The proof makes use of a nonstandard minimax inequality (rather than equality as in the standard case) that is available in the literature. In the second part of the paper, we apply our results in concrete probabilistic settings for systemic risk measures, in particular, in the context of Eisenberg-Noe clearing model. We also provide novel economic interpretations of the dual representations in these settings.

q-fin.RM↗

arXiv2025

Higher-Order Ambiguity Attitudes

We introduce a model-free preference under ambiguity, as a primitive trait of behavior, which we apply once as well as repeatedly. Its single and double application yield simple, easily interpretable definitions of ambiguity aversion and ambiguity prudence. We derive their implications within canonical models for decision under risk and ambiguity. We establish in particular that our new definition of ambiguity prudence is equivalent to a positive third derivative of: (i) the capacity in the Choquet expected utility model, (ii) the dual conjugate of the divergence function under variational divergence preferences and (iii) the ambiguity attitude function in the smooth ambiguity model. We show that our definition of ambiguity prudent behavior may be naturally linked to an optimal insurance problem under ambiguity.

q-fin.RM↗

arXiv2024

On Geometrically Convex Risk Measures

Geometrically convex functions constitute an interesting class of functions obtained by replacing the arithmetic mean with the geometric mean in the definition of convexity. As recently suggested, geometric convexity may be a sensible property for financial risk measures ([7,13,4]). We introduce a notion of GG-convex conjugate, parallel to the classical notion of convex conjugate introduced by Fenchel, and we discuss its properties. We show how GG-convex conjugation can be axiomatized in the spirit of the notion of general duality transforms introduced in [2,3]. We then move to the study of GG-convex risk measures, which are defined as GG-convex functionals defined on suitable spaces of random variables. We derive a general dual representation that extends analogous expressions presented in [4] under the additional assumptions of monotonicity and positive homogeneity. As a prominent example, we study the family of Orlicz risk measures. Finally, we introduce multiplicative versions of the convex and of the increasing convex order and discuss related consistency properties of law-invariant GG-convex risk measures.

q-fin.RM↗

arXiv2023

Elicitability of Return Risk Measures

Informally, a risk measure is said to be elicitable if there exists a suitable scoring function such that minimizing its expected value recovers the risk measure. In this paper, we analyze the elicitability properties of the class of return risk measures (i.e., normalized, monotone and positively homogeneous risk measures). First, we provide dual representation results for convex and geometrically convex return risk measures. Next, we establish new axiomatic characterizations of Orlicz premia (i.e., Luxemburg norms). More specifically, we prove, under different sets of conditions, that Orlicz premia naturally arise as the only elicitable return risk measures. Finally, we provide a general family of strictly consistent scoring functions for Orlicz premia, a myriad of specific examples and a mixture representation suitable for constructing Murphy diagrams.

q-fin.RM↗