Law-invariant BSDEs and dynamic risk measures: new characterizations

<div> 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. </div>