# Towards a Topological Representation of Risks and Their Measures

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

- (a)
- $\varnothing ,X\in \mathcal{R},$ where ∅ is the empty set.
- (b)
- If ${\left\{{K}_{a}\right\}}_{a\in A}$ is a collection of sets such that ${K}_{a}\in \mathcal{R}$ for any $a\in A,$ then also $\bigcup _{a\in A}}{K}_{a}\in \mathcal{R},$ and if ${K}_{1},\dots ,{K}_{n}$ is a finite collection of sets such that ${K}_{j}\in \mathcal{R}$ for any $j=1,2,\dots ,n,$ then $\bigcap _{1\le j\le n}}{K}_{j}\in \mathcal{R$.
- (c)
- For any TR random set X, there exists a TR set-valued measure ϱ that quantifies the amount of loss expected from the risk, and it is defined by $\varrho :X\in \mathcal{R}\u27f9\varrho \left(X\right)\in \mathcal{R}\subseteq {\mathbb{R}}^{d}$ that is a d-dimensional risk measure.
- (d)
- Uncertainty. Any topological risk posses an uncertainty distance $d:X\times \varrho \left(X\right)\to [0,\infty ]$ on X such that $d\left(x,\varrho \left(X\right)\right)$ is a random metric on $\mathcal{R}$ where $x\in X\subseteq {\mathbb{R}}^{d}$ is a d-dimensional random vector on the set X. The distance metric d quantifies the variation of the risk with its risk measure. To quantify the uncertainty measure, we take its expectation $\mathrm{E}\left(d\left(x,\varrho \left(X\right)\right)\right)$.

## 2. Main Results

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

- 1.
- For sets ${A}_{j}\in \mathcal{R}$ and ${B}_{j}\in \mathcal{R},j=0,\dots ,n$, the following additivity property holds$$\Xi \left({\{{A}_{j}\cup {B}_{j}\}}_{j=0}^{m}\right)=\Xi ({\{{A}_{j}\}}_{j=0}^{m})+\Xi \left({\{{B}_{j}\}}_{j=0}^{m}\right)-\Xi \left({\{{A}_{j}\cap {B}_{j}\}}_{j=0}^{m}\right),$$
- 2.
- For empty sets ${X}_{0},{X}_{1},\dots =\varphi $ the portfolio characteristic Ξ is zero.
- 3.
- $\Xi \left({P}_{X}\right)$ can be written as the sum of Betti numbers$$\Xi \left({P}_{X}\right)=\sum _{j=0}^{m}\sum _{k=0}^{dim{M}_{j}}{\left(-1\right)}^{k}{\alpha}_{j}{\beta}_{k,j}.$$

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

## 3. From Topological Risks to $\left(\mathit{\varrho},\mathit{d}\right)$ Risk Space

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

#### Numerical Illustration

## 4. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Shushi, T.
Towards a Topological Representation of Risks and Their Measures. *Risks* **2018**, *6*, 134.
https://doi.org/10.3390/risks6040134

**AMA Style**

Shushi T.
Towards a Topological Representation of Risks and Their Measures. *Risks*. 2018; 6(4):134.
https://doi.org/10.3390/risks6040134

**Chicago/Turabian Style**

Shushi, Tomer.
2018. "Towards a Topological Representation of Risks and Their Measures" *Risks* 6, no. 4: 134.
https://doi.org/10.3390/risks6040134