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The property (4) represents what has become known in the literature as the strong time consistency property.
For example, if (mathcal {Q}={P}), then the strong time consistency reduces to the tower property for conditional expectations.
(i) In general, for dynamic LM-measures, the strong time consistency does not imply either the weak acceptance or weak rejection time consistency.
The first study of dynamic risk measures for stochastic processes (finite probability space and discrete time) is attributed to Riedel (2004), where the author introduced the (strong) time consistency as one of the axioms.
An important, and frequently studied, type of strong time consistency is the strong time consistency for dynamic monetary risk measures on L ∞ (cf. (Acciaio and Penner 2011) and references therein).
Also in (Bielecki et al. 2016), the authors study the strong time consistency of quasi-concave maps via the concept of certainty equivalence; see also (Frittelli and Maggis 2010).
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