Dynamic Return and Star-Shaped Risk Measures via BSDEs

Zullino Marco, University of Pavia and University of Amsterdam

This paper presents characterization results for dynamic return and star-shaped risk measures induced through Backward Stochastic Differential Equations (BSDEs). The paper begins by characterizing any static star-shaped functional as the minimum of convex functionals in a locally convex Fréchet lattice. This generalization extends the findings in Castellani (2000) 'A dual representation for proper positively homogeneous functions' to include star-shaped functions. Additionally, the paper goes beyond the approach proposed in Castagnoli et at. (2022) 'Star-shaped risk measures' by dropping the axioms of cash-additivity, normalization and monotonicity. Next, a regularization procedure is employed to construct a suitable family of convex drivers for BSDEs, which in turn induces a corresponding family of dynamic convex risk measures. The dynamic return and star-shaped risk measures are identified as essential minimums within this family. Furthermore, it is shown that every driver of a star-shaped risk measure induced by BSDEs is star-shaped, establishing a connection to the converse comparison theorem for BSDEs, akin to Jiang (2008) 'Convexity, translation invariance and subadditivity for g-expectations and related risk measures'. Conditions are also established under which the minimum of a family of convex risk measures can be obtained as the solution to a BSDE with a star-shaped driver, utilizing the techniques provided in Peng (2004) 'Nonlinear Expectations, Nonlinear Evaluations and Risk Measures'. The paper further develops the theory of supersolutions for star-shaped drivers and obtains a dual representation, expanding upon the theory developed for convex functionals in Drapeau et al. (2014) 'Dual representation of minimal supersolution of convex BSDEs' to include star-shaped risk measures. Specifically, it is proven that if the set of star-shaped supersolutions for a BSDE is not empty, then there exists a family of convex supersolutions such that at least one element of the family has a non-empty set of supersolutions, yielding the represention of star-shaped minimal supersolutions as the minimum of the family of convex minimal supersolutions. Theoretical results are illustrated through a few examples, showcasing their usefulness in two applications: capital allocation and portfolio choice.

Area: CS55 - New probabilistic approaches in mathematical finance (Lorenzo Torricelli)

Keywords: Backward Stochastic Differential Equations; Dynamic risk measures; Return risk measures; Star-shapedness; Positive homogeneity.

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