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

TL;DR

This paper explores a generalized risk measure framework using two-regime step functions, analyzing optimal thresholds and regime changes for log-concave and elliptic distributions in one and multiple dimensions.

Contribution

It introduces a novel two-regime risk measure model and provides conditions for regime change uniqueness in log-concave distributions, extending classical risk assessment methods.

Findings

Conditions for uniqueness of regime change in log-concave distributions

Extension of risk measures to two-regime step functions

Counterexamples demonstrating the necessity of convexity

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.