On the subdifferential of some critical convex risk functions

<p style="text-align:center;"> <span><strong>(Communicated by Shijing Si)</strong></span> <div style="text-align:justify;"> Risk minimization problems are commonly used in financial and economic decisions, but the solution approach is unmatured even for the convex risk functions. This paper aims at a subdifferential solution approach for three critical classes of convex risk functions: (1) risk functions of composite structure, (2) risk functions with centralized transformation, and (3) risk functions represented by optimal value functions. We analyze the subdifferential properties of the three critical classes of risk functions, which are closely related to the so-called risk quadrangle and covers many commonly used risk functions. We also discuss heir applications on the subgradient method for the risk minimization problems with convex risk functions. </div>