A property of log-concave and weakly-symmetric distributions for two-step approximations of random variables

Source abstract

In this paper, we introduce a generalization of classical risk measures in which the risk is represented by a step function taking two values, corresponding to two endogenously determined market regimes. This extends the traditional framework where risk measures map random variables to single real numbers. For the quadratic loss function, we study the optimization problem of determining the optimal regime threshold and corresponding values. In the case of log-concave distributions, we give conditions for the uniqueness of the regime changing. We treat the case of one dimension and also of multi-dimensions for elliptic distributions. We demonstrate the necessity of convexity through counterexamples.