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

Mihaela-Adriana Nistor, Ionel Popescu

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.

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

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.
Paper Structure (13 sections, 6 theorems, 86 equations, 2 figures)

This paper contains 13 sections, 6 theorems, 86 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The one dimensional case
  3. Problem setup
  4. Continuous and log-concave distributions in one dimension
  5. Examples
  6. Settings and the result.
  7. Regularity assumptions
  8. Some integral inequalities for convex functions
  9. The proof of Theorem \ref{['t:1']}
  10. The multidimensional case
  11. Problem definition and classes of regimes
  12. Considerations on convexity
  13. Counterexample for the halfspace functional (integer vertices)

Key Result

Proposition 1

Assume that $G:\mathbb{R}\to\mathbb{R}$ is convex. Then for any fixed $\alpha,\beta\in\mathbb{R}$, a minimizer of over Borel sets $A\subset\mathbb{R}$ is given (up to $\mathbb{P}$-null sets) by Moreover, $A_{\alpha,\beta}$ is an interval (possibly empty or all of $\mathbb{R}$). In particular, unless $\alpha=\beta$, $A_{\alpha,\beta}$ is a half-line.

Figures (2)

  • Figure 1: Example \ref{['ex:two-maxima-fx']}: a symmetric, non-log-concave density with global maximizers of $f_X$ at $\pm t^\star$, where $t^\star=2\sqrt5-4\approx 0.4721$.
  • Figure 2: Integer-vertex hexagon $K$ and slices $x=0$ (dotted), $x=1$ (dashed).

Theorems & Definitions (19)

  • Proposition 1
  • proof
  • Definition 1
  • Theorem 1
  • Lemma 1
  • proof
  • proof
  • Proposition 2
  • proof
  • Example 1
  • ...and 9 more