  As a counterpart to the (static) risk measures of generalized quantiles and
motivated by Bellini et al. (2018), we propose a new kind of conditional risk
measure called conditional generalized quantiles. We first show their
well-definedness and they can be equivalently characterised by a conditional
first order condition. We also discuss their main properties, and, especially,
We give the characterization of coherency/convexity. For potential applications
as a dynamic risk measure, we study their time consistency properties, and
establish their equivalent characterizations among conditional generalized
quantiles.
