Deviations of convex and coherent entropic risk measures

In this article, we establish several deviations for convex and coherent entropic risk measures. Firstly, we provide several deviations for the two risk measures with respect to relative entropy. Secondly, we provide several deviations for the two risk measures with respect to parameters. Thirdly, we study the continuity properties of the two risk measures with respect to the random variables under several norms. Finally, we establish an application of our main results.