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arXiv · 2604.11382

Law-invariant BSDEs and dynamic risk measures: new characterizations

Zakaria Bensaid·Roxana Dumitrescu·Anis Matoussi·Wissal Sabbagh
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Abstract

We provide a new characterization of law-invariant backward stochastic differential equations (i.e. BSDEs) with quadratic growth. This answers the open question raised in Xu--Xu--Zhou (2022) on necessary conditions for law-invariance of g-expectations, and extends the analysis to general (possibly non-deterministic) generators. We also introduce and compare several dynamic notions of law-invariance in continuous time, establishing precise relationships among them. As an application, we study dynamic risk measures. For cash-additive, normalized risk measures, we recover and extend to continuous time the Kupper--Schachermayer (2009) characterization obtained in discrete time, showing that law-invariance and strong time-consistency force an entropic structure. We further obtain a new characterization of cash non-additive law-invariant risk measures generated by BSDEs via a time-dependent certainty equivalent representation.

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Zakaria Bensaid, Roxana Dumitrescu, Anis Matoussi, Wissal Sabbagh. 2026-04-13. Law-invariant BSDEs and dynamic risk measures: new characterizations. https://arxiv.org/abs/2604.11382

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