# Spatial Risk Measures and Rate of Spatial Diversification

## Abstract

**:**

## 1. Introduction

## 2. Spatial Risk Measures and Other Concepts

#### 2.1. Spatial Risk Measures and Corresponding Axioms

**Definition**

**1**

**.**For $A\in \mathcal{A}$ and $P\in \mathcal{P}$, the normalized spatially aggregated loss is defined by

**Proposition**

**1.**

**Proof.**

**Definition**

**2**

**.**A spatial risk measure is a function ${\mathcal{R}}_{\mathsf{\Pi}}$ that assigns a real number to any region $A\in \mathcal{A}$ and distribution $P\in \mathcal{P}$:

**Definition**

**3**

**.**Let Π be a classical and law-invariant risk measure. For a fixed $P\in \mathcal{P}$, we define the following axioms for the spatial risk measure associated with Π and induced by P, ${\mathcal{R}}_{\mathsf{\Pi}}(\xb7,P)$:

- 1.
- Spatial invariance under translation:for all $\mathbf{v}\in {\mathbb{R}}^{2}$ and $A\in \mathcal{A},\phantom{\rule{4pt}{0ex}}{\mathcal{R}}_{\mathsf{\Pi}}(A+\mathbf{v},P)={\mathcal{R}}_{\mathsf{\Pi}}(A,P)$, where $A+\mathbf{v}$ denotes the region A translated by the vector $\mathbf{v}$.
- 2.
- Spatial sub-additivity:for all ${A}_{1},{A}_{2}\in \mathcal{A},\phantom{\rule{4pt}{0ex}}{\mathcal{R}}_{\mathsf{\Pi}}({A}_{1}\cup {A}_{2},P)\le min\{{\mathcal{R}}_{\mathsf{\Pi}}({A}_{1},P),{\mathcal{R}}_{\mathsf{\Pi}}({A}_{2},P)\}$.
- 3.
- Asymptotic spatial homogeneity of order $-\gamma ,\gamma \ge 0$:for all $A\in {\mathcal{A}}_{c}$,$${\mathcal{R}}_{\mathsf{\Pi}}(\lambda A,P)\underset{\lambda \to \infty}{=}{K}_{1}(A,P)+{\displaystyle \frac{{K}_{2}(A,P)}{{\lambda}^{\gamma}}}+o\left(\frac{1}{{\lambda}^{\gamma}}\right),$$

**Remark**

**1.**

**Theorem**

**1.**

**Proof.**

**Definition**

**4**

**.**The normalized spatially aggregated loss function is defined by

**Definition**

**5**

**.**A spatial risk measure is a function ${\mathcal{R}}_{\mathsf{\Pi}}$ that assigns a real number to any region $A\in \mathcal{A}$ and random field $C\in \mathcal{C}$:

**Definition**

**6**

**.**Let Π be a classical risk measure. For a fixed $C\in \mathcal{C}$, we define the following axioms for the spatial risk measure associated with Π and induced by C, ${\mathcal{R}}_{\mathsf{\Pi}}(\xb7,C)$:

- 1.
- Spatial invariance under translation:for all $\mathbf{v}\in {\mathbb{R}}^{2}$ and $A\in \mathcal{A},\phantom{\rule{4pt}{0ex}}{\mathcal{R}}_{\mathsf{\Pi}}(A+\mathbf{v},C)={\mathcal{R}}_{\mathsf{\Pi}}(A,C)$, where $A+\mathbf{v}$ denotes the region A translated by the vector $\mathbf{v}$.
- 2.
- Spatial sub-additivity:for all ${A}_{1},{A}_{2}\in \mathcal{A},\phantom{\rule{4pt}{0ex}}{\mathcal{R}}_{\mathsf{\Pi}}({A}_{1}\cup {A}_{2},C)\le min\{{\mathcal{R}}_{\mathsf{\Pi}}({A}_{1},C),{\mathcal{R}}_{\mathsf{\Pi}}({A}_{2},C)\}$.
- 3.
- Asymptotic spatial homogeneity of order $-\gamma ,\gamma \ge 0$:for all $A\in {\mathcal{A}}_{c}$,$${\mathcal{R}}_{\mathsf{\Pi}}(\lambda A,C)\underset{\lambda \to \infty}{=}{K}_{1}(A,C)+{\displaystyle \frac{{K}_{2}(A,C)}{{\lambda}^{\gamma}}}+o\left(\frac{1}{{\lambda}^{\gamma}}\right),$$

#### 2.2. Concrete Applications to Insurance

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

#### 2.3. Mixing and Central Limit Theorems for Random Fields

**Theorem**

**2.**

**Proof.**

#### 2.4. Max-Stable Random Fields

**Definition**

**7**

**.**A real-valued random field ${\left\{Z(\mathbf{x})\right\}}_{\mathbf{x}\in {\mathbb{R}}^{d}}$ is said to be max-stable if there exist sequences of functions ${({a}_{T}(\mathbf{x}),\mathbf{x}\in {\mathbb{R}}^{d})}_{T\ge 1}>0$ and ${({b}_{T}(\mathbf{x}),\mathbf{x}\in {\mathbb{R}}^{d})}_{T\ge 1}\in \mathbb{R}$ such that, for all $T\ge 1$,

**Definition**

**8**

**.**Let ${\{W(\mathbf{x})\}}_{\mathbf{x}\in {\mathbb{R}}^{d}}$ be a centred Gaussian random field with stationary increments and with variogram ${\gamma}_{W}$. Let us consider the random field Y defined by

**Definition**

**9**

**.**Let Z be written as in (18) with f being the density of a d-variate Gaussian random vector with mean $\mathbf{0}$ and covariance matrix Σ . Then, the field Z is referred to as the Smith random field with covariance matrix Σ .

## 3. Properties of Some Induced Spatial Risk Measures

#### 3.1. General Cost Field

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

- 1.
- For any $A\in \mathcal{A}$,$$\underset{\mathbf{x}\in A}{sup}\left\{\mathbb{E}\left[{[C(\mathbf{x})]}^{2}\right]\right\}<\infty .$$
- 2.
- For all $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^{2}$,$$\mathrm{Cov}(C(\mathbf{x}),C(\mathbf{y}))=\mathrm{Cov}(C(\mathbf{0}),C(\mathbf{x}-\mathbf{y})),$$$${\int}_{{\mathbb{R}}^{2}}\left|\mathrm{Cov}(C(\mathbf{0}),C(\mathbf{x}))\right|\phantom{\rule{4pt}{0ex}}\nu (\mathrm{d}\mathbf{x})<\infty .$$

**Proof.**

**Theorem**

**5.**

- 1.
- Assume that C is stationary. Then, provided it exists, any spatial risk measure associated with a law-invariant classical risk measure Π and induced by C satisfies the axiom of spatial invariance under translation.
- 2.
- Assume that C is such that, for all $\mathbf{x}\in {\mathbb{R}}^{2}$,$$\mathbb{E}\left[{[C(\mathbf{x})]}^{2}\right]<\infty ,$$$${\mathcal{R}}_{2}(\lambda A,C)\underset{\lambda \to \infty}{=}{\displaystyle \frac{{\sigma}_{C}^{2}}{{\lambda}^{2}\nu (A)}}+o\left({\displaystyle \frac{1}{{\lambda}^{2}}}\right).$$Hence, if ${\sigma}_{C}>0$, ${\mathcal{R}}_{2}(\xb7,C)$ satisfies the axiom of asymptotic spatial homogeneity of order $-2$ with ${K}_{1}(A,C)=0$ and ${K}_{2}(A,C)={\sigma}_{C}^{2}/\nu (A),A\in {\mathcal{A}}_{c}$.
- 3.
- Assume that C has a constant expectation and satisfies the CLT. Then, we have, for all $A\in {\mathcal{A}}_{c}$, that$${\mathcal{R}}_{3,\alpha}(\lambda A,C)\underset{\lambda \to \infty}{=}\mathbb{E}[C(\mathbf{0})]+\frac{{\sigma}_{C}{q}_{\alpha}}{\lambda {[\nu (A)]}^{\frac{1}{2}}}+o\left(\frac{1}{\lambda}\right).$$Hence, if $\alpha \in (0,1)\backslash \{1/2\}$, ${\mathcal{R}}_{3,\alpha}(\xb7,C)$ satisfies the axiom of asymptotic spatial homogeneity of order $-1$ with ${K}_{1}(A,C)=\mathbb{E}[C(\mathbf{0})]$ and ${K}_{2}(A,C)={\sigma}_{C}{q}_{\alpha}/{[\nu (A)]}^{\frac{1}{2}},A\in {\mathcal{A}}_{c}$.
- 4.
- Assume that C has a constant expectation, satisfies the CLT and is such that the random variables $\lambda \left({L}_{N}(\lambda A,C)-\mathbb{E}[C(\mathbf{0})]\right)$, $\lambda >0$, are uniformly integrable. Then, we have, for all $A\in {\mathcal{A}}_{c}$, that$${\mathcal{R}}_{4,\alpha}(\lambda A,C)\underset{\lambda \to \infty}{=}\mathbb{E}[C(\mathbf{0})]+\frac{{\sigma}_{C}\varphi ({q}_{\alpha})}{\lambda {[\nu (A)]}^{\frac{1}{2}}(1-\alpha )}+o\left(\frac{1}{\lambda}\right).$$Hence, ${\mathcal{R}}_{4,\alpha}(\xb7,C)$ satisfies the axiom of asymptotic spatial homogeneity of order $-1$ with ${K}_{1}(A,C)=\mathbb{E}[C(\mathbf{0})]$ and ${K}_{2}(A,C)={\sigma}_{C}\varphi ({q}_{\alpha})/\{{[\nu (A)]}^{\frac{1}{2}}(1-\alpha )\}$, $A\in {\mathcal{A}}_{c}$.

**Proof.**

**Remark**

**5.**

**Corollary**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

#### 3.2. Cost Field Being a Function of a Max-Stable Random Field

**Corollary**

**2.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

- 1.
- ${\mathcal{R}}_{2}(\xb7,C)$ satisfies the axiom of asymptotic spatial homogeneity of order $-2$ with ${K}_{1}(A,C)=0$ and ${K}_{2}(A,C)={\sigma}_{C}^{2}/\nu (A),A\in {\mathcal{A}}_{c}$.
- 2.
- For all $\alpha \in (0,1)\backslash \{1/2\}$, ${\mathcal{R}}_{3,\alpha}(\xb7,C)$ satisfies the axiom of asymptotic spatial homogeneity of order $-1$ with ${K}_{1}(A,C)=\mathbb{E}[C(\mathbf{0})]$ and ${K}_{2}(A,C)={\sigma}_{C}{q}_{\alpha}/{[\nu (A)]}^{\frac{1}{2}}$, $A\in {\mathcal{A}}_{c}$.
- 3.
- For all $\alpha \in (0,1)$, ${\mathcal{R}}_{4,\alpha}(\xb7,C)$ satisfies the axiom of asymptotic spatial homogeneity of order $-1$ with ${K}_{1}(A,C)=\mathbb{E}[C(\mathbf{0})]$ and ${K}_{2}(A,C)={\sigma}_{C}\varphi ({q}_{\alpha})/\{{[\nu (A)]}^{\frac{1}{2}}(1-\alpha )\}$, $A\in {\mathcal{A}}_{c}$.

**Proof.**

**Corollary**

**3.**

- 1.
- ${\mathcal{R}}_{2}(\xb7,C)$ satisfies the axiom of asymptotic spatial homogeneity of order $-2$ with ${K}_{1}(A,C)=0$ and ${K}_{2}(A,C)={\sigma}_{C}^{2}/\nu (A),A\in {\mathcal{A}}_{c}$.
- 2.
- For all $\alpha \in (0,1)\backslash \{1/2\}$, ${\mathcal{R}}_{3,\alpha}(\xb7,C)$ satisfies the axiom of asymptotic spatial homogeneity of order $-1$ with ${K}_{1}(A,C)=\mathbb{E}[C(\mathbf{0})]$ and ${K}_{2}(A,C)={\sigma}_{C}{q}_{\alpha}/{[\nu (A)]}^{\frac{1}{2}}$, $A\in {\mathcal{A}}_{c}$.
- 3.
- For all $\alpha \in (0,1)$, ${\mathcal{R}}_{4,\alpha}(\xb7,C)$ satisfies the axiom of asymptotic spatial homogeneity of order $-1$ with ${K}_{1}(A,C)=\mathbb{E}[C(\mathbf{0})]$ and ${K}_{2}(A,C)={\sigma}_{C}\varphi ({q}_{\alpha})/\{{[\nu (A)]}^{\frac{1}{2}}(1-\alpha )\}$, $A\in {\mathcal{A}}_{c}$.

**Proof.**

**Theorem**

**8.**

- 1.
- ${\mathcal{R}}_{2}(\xb7,C)$ satisfies the axiom of asymptotic spatial homogeneity of order $-2$ with ${K}_{1}(A,C)=0$ and ${K}_{2}(A,C)={\sigma}_{C}^{2}/\nu (A),A\in {\mathcal{A}}_{c}$.
- 2.
- For all $\alpha \in (0,1)\backslash \{1/2\}$, ${\mathcal{R}}_{3,\alpha}(\xb7,C)$ satisfies the axiom of asymptotic spatial homogeneity of order $-1$ with ${K}_{1}(A,C)=\mathbb{E}[C(\mathbf{0})]$ and ${K}_{2}(A,C)={\sigma}_{C}{q}_{\alpha}/{[\nu (A)]}^{\frac{1}{2}}$, $A\in {\mathcal{A}}_{c}$.
- 3.
- For all $\alpha \in (0,1)$, ${\mathcal{R}}_{4,\alpha}(\xb7,C)$ satisfies the axiom of asymptotic spatial homogeneity of order $-1$ with ${K}_{1}(A,C)=\mathbb{E}[C(\mathbf{0})]$ and ${K}_{2}(A,C)={\sigma}_{C}\varphi ({q}_{\alpha})/\{{[\nu (A)]}^{\frac{1}{2}}(1-\alpha )\}$, $A\in {\mathcal{A}}_{c}$.

**Proof.**

**Corollary**

**4.**

- 1.
- 2.
- 3.

**Proof.**

**Theorem**

**9.**

- 1.
- 2.
- 3.

**Proof.**

**Corollary**

**5.**

- 1.
- 2.
- 3.

**Proof.**

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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1 | Throughout the paper, insurance also refers to reinsurance. |

2 | Throughout the paper, when applied to random fields, the adjective “measurable” means “jointly measurable”. |

3 | Unless otherwise stated, by a.s., we mean $\mathbb{P}$-a.s. |

4 | See Section 2.2. |

5 | Throughout the paper, stationarity refers to strict stationarity. |

6 | It is out of the scope of this paper to enter into accounting details. |

7 | Potentially different from those developed in the natural catastrophes industry: e.g., a max-stable model. |

8 | In the following, when W is sample-continuous, what we refer to as the Brown–Resnick random field built with W is obtained by taking replications of W (see (17)) which are also sample-continuous. |

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**MDPI and ACS Style**

Koch, E.
Spatial Risk Measures and Rate of Spatial Diversification. *Risks* **2019**, *7*, 52.
https://doi.org/10.3390/risks7020052

**AMA Style**

Koch E.
Spatial Risk Measures and Rate of Spatial Diversification. *Risks*. 2019; 7(2):52.
https://doi.org/10.3390/risks7020052

**Chicago/Turabian Style**

Koch, Erwan.
2019. "Spatial Risk Measures and Rate of Spatial Diversification" *Risks* 7, no. 2: 52.
https://doi.org/10.3390/risks7020052