The aim of this work is to study risk measures generated by distortion
functions in a dynamic discrete time setup, and to investigate the
corresponding dynamic coherent acceptability indices (DCAIs) generated by
families of such risk measures. First we show that conditional version of
Choquet integrals indeed are dynamic coherent risk measures (DCRMs), and also
introduce the class of dynamic weighted value at risk measures. We prove that
these two classes of risk measures coincides. In the spirit of robust
representations theorem for DCAIs, we establish some relevant properties of
families of DCRMs generated by distortion functions, and then define and study
the corresponding DCAIs. Second, we study the time consistency of DCRMs and
DCAIs generated by distortion functions. In particular, we prove that such
DCRMs are sub-martingale time consistent, but they are not super-martingale
time consistent. We also show that DCRMs generated by distortion functions are
not weakly acceptance time consistent. We also present several widely used
classes of distortion functions and derive some new representations of these
distortions.Comment: This manuscript is accompanied by a supplement that contains some
technical, but important, results and their proof