## Geometrically convex risk measures

Geometrically convex functions, also known as multiplicatively convex functions or GG-convex functions (see e.g. Niculescu, 2000) are an interesting class of functions defined by replacing the geometric mean to the arithmetic mean in the definition of convexity. As discussed in the financial literature, geometric convexity might be a sensible property for risk measures, in particular if they are applied to financial returns (see Bellini et al., 2018, Laeven and Rosazza Gianin, 2022, Aygun et al., 2023). Our first contribution is to introduce a notion of geometrically convex conjugate for real functions, similar to the usual notion of Fenchel transform, and to study its properties and its axiomatic foundation, in the spirit of Artstein-Avidan and Milman (2009). As a consequence, we derive dual representations for geometrically convex risk measures, related to the ones that can be found in Laeven and Rosazza Gianin, 2022 and Aygun et al., 2023. We study in detail risk measures based on Orlicz norms, that are an interesting family illustrating the relevance of the various notions of convexity. Finally, we study the properties of a naturally defined geometrically convex order among real functions, and discuss the consistency properties of risk measures with respect to this stochastic order. \\ Artstein-Avidan, S., Milman, V. (2009). The concept of duality in convex analysis, and the characterization of the Legendre transform. Annals of Mathematics 169, 661-674.\\ Aygün, M., Bellini, F., Laeven, R.J.A.(2023). Elicitability of Return Risk Measures. Working paper, https://arxiv.org/abs/2302.13070v2\\ Bellini, F., Laeven, R.J.A., Rosazza Gianin, E. (2018). Robust Return Risk Measures. Mathematics and Financial Economics 12(1), 5-32\\ Laeven, R., Rosazza Gianin, E. (2022). Quasi-Logconvex measures of risk. Working paper, https://arxiv.org/abs/2208.07694\\ Niculescu, C.P.(2000). Convexity according to the geometric mean. Math. Inequal. Appl. 2, 155-167.

Area: CS20 - Stochastic Modeling in Finance and Insurance I (Tiziano de Angelis)

Keywords: Geometric convexity, Risk Measures, Duality, Orlicz premia, Stochastic Orders.

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