[Paper Review] Concave Distortion Semigroups

Alexander S. Cherny, Damir Filipović|arXiv (Cornell University)|Apr 4, 2011

Probabilistic and Robust Engineering Design8 references3 citations

TL;DR

This paper introduces concave distortion semigroups—families of concave, increasing functions $\Psi_t: [0,1] \to [0,1]$ satisfying the semigroup property $\Psi_s \circ \Psi_t = \Psi_{s+t}$—to narrow the class of coherent acceptability indices. It establishes a one-to-one correspondence between such semigroups and concave generators $G: (0,1) \to (0,\infty)$, enabling unique reconstruction of the full semigroup from a single distortion $\Psi_{t_*}$ under regularity conditions, and identifies canonical examples like the AIMAX and Wang transform semigroups.

ABSTRACT

The problem behind this paper is the proper measurement of the degree of quality/acceptability/distance to arbitrage of trades. We are narrowing the class of coherent acceptability indices introduced by Cherny and Madan (2007) by imposing an additional mathematical property. For this, we introduce the notion of a concave distortion semigroup as a family $(Ψ_t)_{t\ge0}$ of concave increasing functions $[0,1] o[0,1]$ satisfying the semigroup property $$ Ψ_s\circΨ_t=Ψ_{s+t},\quad s,t\ge0. $$ The goal of the paper is the investigation of these semigroups with regard to the following aspects: representation of distortion semigroups; properties of distortion semigroups desirable from the economical or mathematical perspective; determining which concave distortions belong to some distortion semigroup.

Motivation & Objective

• To narrow the broad class of coherent acceptability indices by introducing the semigroup property as a mathematical constraint.
• To identify a canonical subclass of Weighted V@R acceptability indices via concave distortion semigroups.
• To address the practical problem of recovering a full family of distortions $\Psi_t$ from a single observed distortion $\Psi_{t_*}$ estimated from market data.
• To characterize the conditions under which such recovery is unique and to provide an approximation algorithm for the generator.
• To explore closure properties of the resulting class of indices under operations like maximum and infimum.

Proposed method

• Define a concave distortion semigroup as a family $(\Psi_t)_{t \geq 0}$ of concave, increasing functions $[0,1] \to [0,1]$ satisfying $\Psi_s \circ \Psi_t = \Psi_{s+t}$.
• Establish a one-to-one correspondence between such semigroups and concave generators $G: (0,1) \to (0,\infty)$ via the integral equation $\Psi_t(x) = \inf\{ y \in [x,1] : \int_x^y \frac{1}{G(s)} ds = t \}$.
• Define the generator as $G(x) = \lim_{t \downarrow 0} \frac{\Psi_t(x) - x}{t}$, linking infinitesimal behavior to the semigroup.
• Prove that for any given concave distortion $\Psi$ with $\Psi'_-(1) > 0$, there exists a unique concave distortion semigroup such that $\Psi_1 = \Psi$, enabling data-driven recovery.
• Provide an iterative approximation algorithm for computing the generator $G$ from a given $\Psi_{t_*}$.
• Analyze closure properties: the infimum and maximum of two semigroup indices correspond to the pointwise infimum and smallest concave majorant of their generators, respectively.

Experimental results

Research questions

• RQ1 Can a unique concave distortion semigroup be reconstructed from a single observed distortion $\Psi_{t_*}$, and under what conditions?
• RQ2 What mathematical and economic properties characterize the most desirable distortion semigroups within the broader class?
• RQ3 How do operations like infimum and maximum on acceptability indices correspond to operations on their underlying generators?
• RQ4 Which concave distortion functions $\Psi$ admit a generating semigroup, and what is the structure of the set of such functions?
• RQ5 What are the canonical examples of distortion semigroups with optimal properties, and how do they relate to known performance measures like Wang transform or AIMAX?

Key findings

• There is a one-to-one correspondence between concave distortion semigroups $(\Psi_t)_{t \geq 0}$ and concave generators $G: (0,1) \to (0,\infty)$, given by $\Psi_t(x) = \inf\{ y \in [x,1] : \int_x^y \frac{1}{G(s)} ds = t \}$.
• The generator $G(x)$ is recovered as the infinitesimal rate of change: $G(x) = \lim_{t \downarrow 0} \frac{\Psi_t(x) - x}{t}$.
• For any concave distortion $\Psi$ with $\Psi'_-(1) > 0$, there exists a unique concave distortion semigroup such that $\Psi_1 = \Psi$, enabling unique recovery from a single observation.
• An iterative algorithm is provided to approximate the generator $G$ from a given $\Psi_{t_*}$, ensuring practical applicability.
• The class of semigroup indices is closed under the infimum and maximum operations: the infimum of two indices corresponds to the pointwise infimum of their semigroups, and the maximum corresponds to the smallest concave majorant of their generators.
• The AIMAX semigroup and the Wang transform are identified as canonical examples with desirable mathematical and economic properties, lying at the intersection of generality and tractability.