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The aim of this work is to study risk measures generated by distortionfunctions in a dynamic discrete time setup, and to investigate thecorresponding dynamic coherent acceptability indices (DCAIs) generated byfamilies of such risk measures. First we show that conditional version ofChoquet integrals indeed are dynamic coherent risk measures (DCRMs), and alsointroduce the class of dynamic weighted value at risk measures. We prove thatthese two classes of risk measures coincides. In the spirit of robustrepresentations theorem for DCAIs, we establish some relevant properties offamilies of DCRMs generated by distortion functions, and then define and studythe corresponding DCAIs. Second, we study the time consistency of DCRMs andDCAIs generated by distortion functions. In particular, we prove that suchDCRMs are sub-martingale time consistent, but they are not super-martingaletime consistent. We also show that DCRMs generated by distortion functions arenot weakly acceptance time consistent. We also present several widely usedclasses of distortion functions and derive some new representations of thesedistortions.

Time consistency of dynamic risk measures and dynamic performance measures generated by distortion functions