2. Preliminaries

We assume that all random variables considered in this paper are defined on an atomless probability space $(\Omega, \mathcal F, \mathbb{P})$ , ensuring all considered distributions exist. Furthermore, we use $L^p$ for $p \geqslant 1 $ to denote the set of all integrable random variables on the space $(\Omega, \mathcal F, \Pr)$ with respect to $L^p$ -norm.

We represent a loss random variable of an insurance or investment portfolio by $X \in L^p$ where a positive value of $X$  indicates a loss and a negative value indicates a profit. A random variable $X$  following a given distribution $F$  is denoted by $X^F$ . We use $U$  to denote a uniform random variable on (0,1). It is well known that $X^F$ and $ F^{-1}(U)$ follow the same distribution $F$ , denoted by $ X^F \buildrel \mathrm{d} \over = F^{-1}(U)$ , where $F^{-1}(u)=\inf\{x\,:\, F(x) \geqslant u\}$ for any $0 \lt u \lt 1$ is the (left-continuous) quantile function of $F$ .

For any set $ A \in \mathcal F $ , we denote the corresponding indicator function by $\mathbb{I}_A$ . For $x,y \in \mathbb{R}$ , we define $ (x)_+ = \max \{x, 0 \} $ , $x \vee y =\max\{x, y\}$ , and $x\wedge y =\min\{x,y\}$ .

Definition 2.1. A function $g\,:\,[0,1]\mapsto[0,1]$ is said to be a distortion function if it is non-decreasing satisfying $g(0)=0$ and $g(1)=1$ . The distortion risk measure of a random variable $Y$  induced by a distortion function g is defined as

(2.1) \begin{equation}\rho^g(Y)= \int_0^{+\infty} g(1-G_Y(x))\mathrm{d} x -\int_{-\infty}^0 ( 1-g(1-G_Y(x))) \mathrm{d} x,\end{equation}

provided that at least one of the two integrals in (2.1) is finite, where $G_Y$ denotes the distribution function of $Y$ .

In this paper, we assume that the distortion function $g$  is absolutely continuous, with a weight function $\gamma(u)=\partial_-g(x)|_{x=1-u}$ for $0\lt u\lt 1$ , satisfying that $\int_0^1 \gamma(u) \mathrm{d} u=1$ , where $\partial_-$ denotes the derivative from the left. Thus, the distortion risk measure $\rho^g$ of $X^F$ has an alternative representation:

(2.2) \begin{equation} \rho^g(X^F)=\int_0^1\gamma(u)F^{-1}(u)\mathrm{d} u.\end{equation}

If $\gamma(u)$ is increasing, a distortion risk measure with expression (2.2) is a coherent risk measure and is referred to as a spectral risk measure (Acerbi, Reference Acerbi2002). In this paper, we make the following assumption about the weight function $\gamma$ :

Definition 2.2. Let $ k \geqslant 1 $ be an integer. For two distributions $F$  and $G$ , the Wasserstein distance of order $k$ between $F$  and $G$  is given by

\begin{align*}W_k(F, \, G)=W_k(F^{-1}, \, G^{-1}) =\Big (\int_{0}^{1}\left|F^{-1}(x)-G^{-1}(x)\right|^k \mathrm{d} x\Big )^{1/k}, \end{align*}

where $ F^{-1} $ and $ G^{-1}$ are the quantile functions of $F$  and $G$ , respectively.

According to the definition, the $k$ -order Wasserstein distance between two distributions $F$  and $G$  is determined by their quantile functions $F^{-1}$ and $G^{-1}$ . Since a distribution and its quantile function are in one-to-one correspondence, we use $W_k(F, \, G)$ or $W_k(F^{-1}, \, G^{-1}) $ interchangeably to denote the Wasserstein distance between two distributions $F$  and $G$ . In addition, the quantile function of a worst-case distribution is called the worst-case quantile function.

Assume that a decision-maker has a reference distribution $\hat{F}$ for a risk $X$ , and all possible distributions for $X$ belong to the uncertainty set $\mathcal S ( \hat{F};\, k, \varepsilon)$ . Correspondingly, for the reference distribution $\hat{F}$ or the reference quantile function $\hat{F}^{-1}$ , we define the uncertainty set of quantile functions for $X$ as

(2.3) \begin{align} \mathcal Q( \hat{F}^{-1};\, k, \varepsilon) & = \big \{F^{-1}\,:\,F\in \mathcal S ( \hat{F};\, k, \varepsilon) \big \} =\big \{ F^{-1}\,:\, W_k ( F^{-1}, \hat{F}^{-1} ) \leqslant \varepsilon \big \}, \,\,\text{where } k \geqslant 1 \text{ and } \varepsilon \geqslant 0.\end{align}

Thus, $F \in \mathcal S ( \hat{F};\, k, \varepsilon) $ if and only if $F^{-1} \in \mathcal Q( \hat{F}^{-1};\, k, \varepsilon)$ .

3. Worst-case distortion risk measures and worst-case distributions

If $\ell (x) = x $ , then problem (1.4) reduces to $ \sup_{F \in {\mathcal S }(\hat{F};\, k, \varepsilon)} \rho(X^F)$ , which was solved in Liu et al. (Reference Liu, Mao, Wang and Wei2022) for the case where $\rho$ is a coherent distortion risk measure. However, the results and proofs from Liu et al. (Reference Liu, Mao, Wang and Wei2022) cannot be directly applied to problem (1.4) when $\ell$ is a non-degenerated stop-loss, limited loss, or limited stop-loss transform.

In this section, we address problem (1.4) for $\ell= \ell_d$ , $\ell^m$ , and $\ell_d^m$ . To this end, we reformulate problem (1.4) with $\ell= \ell_d$ or $\ell^m$ as a sup-sup or sup-inf problem, thereby reducing it to a one-variable maximization problem. For the case when $\ell= \ell_d^m$ , we can reformulate the problem and solve it using the solutions obtained for $\ell= \ell^m$ .

3.1. Stop-loss transform

In this subsection, we solve problem (1.4) when $\ell= \ell_d$ and $\rho$ is the distortion risk measure $ \rho^g$ , as represented in (2.2). Specifically, we solve the following optimization problem:

(3.1) \begin{align}\sup_{F \in {\mathcal S }(\hat{F};\, k, \varepsilon)} \rho((X^F-d)_+) = \sup\big \{\rho^g\left ( ( X^F-d)_+\right)\,:\, F \in {\mathcal S} (\hat{F};\, k, \varepsilon)\big \}.\end{align}

To avoid discussing the trivial cases, we assume $ \mathrm{ess\mbox{-}inf} X^{\hat{F}} \lt d \lt \mathrm{ess\mbox{-}sup} X^{\hat{F}}$ . For any distribution $F \in {\mathcal S }(\hat{F};\, k, \varepsilon) $ , by applying arguments similar to those in Lemma 4.1 of Cai et al. (Reference Cai, Liu and Yin2024), we have

(3.2) \begin{align}\rho \left ( ( X^{F} - d)_+ \right ) &= \int_{F(d)}^1 \gamma(u) \left (F^{-1}(u) -d \right ) \mathrm{d} u =\max_{\beta \in [0,1]} \int_0^1 \gamma_{1,\beta}(u) \left (F^{-1}(u) -d \right ) \mathrm{d} u,\end{align}

where, for any given $\beta \in [0, 1]$ and the weight function $\gamma$ of the distortion function $g$ , we define

\begin{align*}\gamma_{1,\beta} (u) = \gamma (u) \cdot \mathbb{I}_{[\beta,1]} (u) , \,\,\,0 \lt u \lt 1 .\end{align*}

Note that the weight function $\gamma$ itself is non-negative, with $\Vert \gamma \Vert_1 = \int_0^1 \gamma(u) \mathrm{d} u = g(1)= 1 $ . Under Assumption 2.1, $\gamma $ is non-decreasing on (0,1). Therefore, there exists $ \delta \in (0,1) $ such that $\gamma \gt 0 $ holds on the interval $ ( \delta, 1 ) $ . Consequently, $\gamma_{1,\beta} $ is non-negative, with $ \Vert \gamma_{1,\beta }\Vert_1 \geqslant \int_{\beta \vee \delta}^1 \, \gamma (u) \, \mathrm{d} u \gt 0 $ for all $ \beta\lt 1$ . Following (3.2), the worst-case risk measure value in (3.1) can be written as

(3.3) \begin{align}\sup_{F \in \mathcal S (\hat{F};\, k, \varepsilon)} \rho \left ( ( X^{F} - d)_+ \right ) & = \sup_{F\in \mathcal S(\hat{F};\, k, \varepsilon)} \, \max_{\beta \in [0,1]} \int_\beta^1 \gamma(u) \left (F^{-1}(u) -d \right ) \mathrm{d} u \nonumber \\& = \sup_{\beta \in [0,1 ]} \, \sup_{F\in \mathcal S(\hat{F};\, k, \varepsilon)} \int_\beta^1 \gamma(u) \left (F^{-1}(u) -d \right ) \mathrm{d} u.\end{align}

We can first solve the inner maximization problem of (3.3)

(3.4) \begin{align} H(\beta) \triangleq \sup_{F\in \mathcal S(\hat{F};\, k, \varepsilon)} \int_\beta^1 \gamma(u) \left (F^{-1}(u) -d \right ) \mathrm{d} u, \,\,\,0 \leqslant \beta \leqslant 1 \end{align}

by using Proposition 4 of Liu et al. (Reference Liu, Mao, Wang and Wei2022). Then, we reduce problem (3.1) to a feasible one-variable maximization problem. The following theorem characterizes the worst-case risk measure value and the worst-case distribution, if any, for the problem (3.1).

Theorem 3.1. Suppose that Assumption 2.1 holds and $ \mathrm{ess\mbox{-}inf} X^{\hat{F}} \lt d \lt \mathrm{ess\mbox{-}sup} X^{\hat{F}}$ in the problem (3.1).

1. (i) The worst-case distortion risk measure for the problem (3.1) is given by

   (3.5) \begin{align} \sup_{F \in {\mathcal S }(\hat{F};\, k, \varepsilon)} \rho\left ( (X^F-d)_+\right ) = \max_{\beta \in [0, \, 1]} H(\beta),\end{align}
   where $ H(\beta) $ defined in (3.4) is a continuous function of $\beta \in [0, 1]$ and has the following expression:
   (3.6) \begin{align} H(\beta) = \int_0^1 \gamma_{1,\beta} (u) \, \hat{F}^{-1}(u) \, \mathrm{d} u + \varepsilon \, \Vert \gamma_{1,\beta} \Vert_{\bar{k}} -d\, \Vert \gamma_{1,\beta} \Vert_1, \,\,\,0 \leqslant \beta \leqslant 1. \end{align}

2. (ii) For $k \gt 1 $ , if problem (3.1) has a maximizer $ F^* \in \mathcal S(\hat{F};\, k, \varepsilon) $ , then the quantile function of $F^*$ is given by

   \begin{align*}(F^*)^{-1} (u) = \hat{F}^{-1}(u) + \varepsilon \, \frac{ \left (\gamma_{1,F^*(d)}(u) \right )^{\bar{k}-1}}{\Vert \gamma_{1,F^*(d)} \Vert^{\bar{k}/ k }_{\bar{k}}}, \,\,\,0 \lt u \lt 1 ,\end{align*}
   and $F^*(d)$ is a maximizer of the problem (3.5).

3. (iii) For $ k \gt 1 $ , let $\beta^*= \mathrm{arg\,max}_{\beta \in [0, \, 1]} H(\beta)$ and

   (3.7) \begin{equation} (F_{\beta^*})^{-1} (u) = \hat{F}^{-1}(u) + \varepsilon \, \frac{ \left (\gamma_{1,\beta^*}(u) \right )^{\bar{k}-1}}{\Vert \gamma_{1,\beta^*} \Vert^{\bar{k}/ k }_{\bar{k}}}, \,\,\,0 \lt u \lt 1 .\end{equation}
   If $F_{\beta^*} (d{-}) \leqslant \beta^* \leqslant F_{\beta^*} (d) $ , then $F_{\beta^*}$ is a maximizer of the problem (3.1).

Proof. The proof of this theorem is given in the appendix.

Remark 3.1. The function $H(\beta)$ defined in (3.6) is continuous on [0, 1]. Therefore, there always exists a $\beta^* \in [0, 1]$ such that $\sup_{F \in {\mathcal S }(\hat{F};\, k, \varepsilon)} \rho\left ( (X^F-d)_+\right ) =H(\beta^*)$ . In Theorem 3.1, we first conclude that the supremum of problem (3.1) is finite. Then, we show the form a worst-case distribution must take if such a distribution exists. Finally, we provide sufficient conditions under which a worst-case distribution – or equivalently, a worst-case quantile function – for the problem (3.1) exists.

Furthermore, Theorem 3.1(ii) indicates that if the problem (3.1) has a worst-case distribution $ F^* \in \mathcal S(\hat{F};\, k, \varepsilon) $ , then it holds that $ (F^*)^{-1}(u) \geqslant \hat{F}^{-1}(u)$ for any $0\lt u\lt 1$ . This implies that $\Pr(X^{F^*}\gt x) \geqslant \Pr(X^{\hat{F}} \gt x)$ for all $x$ , meaning that the worst-case distribution is riskier than the reference distribution in the sense of the first-order stochastic dominance (FSD). Additionally, if the reference distribution $\hat{F}$ is a non-negative distribution, that is, $\hat{F}(0{-})=\Pr(X^{\hat{F}} \lt 0)= 0$ , then the worst-case distribution $ F^* $ for the problem (3.1) is also non-negative.

In the context of insurance, where the insurance loss $X$ (such as the amount of a claim or the number of claims) is typically non-negative, it is natural for the insurer or reinsurer to use a non-negative distribution as a reference and to consider only non-negative distributions within a Wasserstein ball.

Specifically, the insurer would be interested in the following subset of the Wasserstein ball:

(3.8) \begin{equation} {\mathcal S}^+ (\hat{F};\, k, \varepsilon) \triangleq \{F\,:\, F \in {\mathcal S} (\hat{F};\, k, \varepsilon) \ \mbox{and} \ F(0{-})=0\}.\end{equation}

Thus, Theorem 3.1 implies that if the reference quantile $\hat{F}^{-1}$ is non-negative and if problem (3.1) has a worst-case distribution $F^*$ , then $(F^*)^{-1}$ is non-negative and also serves as a worst-case distribution $F^*$ for $\sup_{F \in {\mathcal S}^+ (\hat{F};\, k, \varepsilon)} \rho ( ( X^F-d)_+ ).$

In the following example, as applications of Theorem 3.1, we give the worst-case expected shortfall or the worst-case TVaR of a stop-loss transform over a $k$ -order Wasserstein ball.

Example 3.1. Let $\rho = \mathrm{ES}_\alpha$ or $\mathrm{TVaR}_\alpha$ with $ 0 \lt \alpha \lt 1$ be the expected shortfall or TVaR, that is, $\rho (X^F ) =\mathrm{ES}_\alpha(X^F) = \mathrm{TVaR}_\alpha(X^F) = \frac{1}{1-\alpha} \int_\alpha^1 F^{-1} (u) \mathrm{d} u. $ It is well known that the  ES is a coherent distortion risk measure with distortion function $g(t) = \min\{ \frac{1}{1-\alpha} \, t , 1 \} $ , $0 \lt t \lt 1$ , and the corresponding weight function is $\gamma (t) = \frac{1}{1-\alpha} \, \mathbb{I}_{ [\alpha , 1 ] }(t) $ , $0 \lt t \lt 1$ . Consequently, for any $0 \lt \beta \lt 1$ , we have

\begin{align*} \gamma_{1,\beta } (t) = \frac{1}{1-\alpha} \mathbb{I}_{ [\alpha \vee \beta , 1 ] }(t), \,\, 0 \lt t \lt 1, \,\, \text{ with } \ \Vert \gamma_{1,\beta} \Vert_1 = \frac{1 - \alpha \vee \beta}{1-\alpha} \ \text{ and } \ \Vert \gamma_{1,\beta} \Vert_{\bar{k}} = \frac{(1 - \alpha \vee \beta)^{ 1 / \bar{k}}}{1-\alpha}. \end{align*}

Then (3.5) in Theorem 3.1 implies

(3.9) \begin{align}\sup_{F \in \mathcal S(\hat{F};\, k, \varepsilon) } \mathrm{ES}_\alpha \left ( ( X^{F} -d)_+ \right )&= \max_{\beta \in [0,1]} \Big \{\frac{1}{1-\alpha} \int_{\alpha \vee \beta }^1 \hat{F}^{-1}(u) \mathrm{d} u + \varepsilon \cdot \frac{(1 - \alpha \vee \beta)^{ 1 / \bar{k}}}{1-\alpha} -d\cdot \frac{1 - \alpha \vee \beta}{1-\alpha} \Big \} \nonumber\\&= \max_{\beta \in [0,1]} \Big \{\frac{1}{1-\alpha} \int_{\alpha \vee \beta }^1 \left ( \hat{F}^{-1}(u) -d \right )\mathrm{d} u + \varepsilon \cdot \frac{(1 - \alpha \vee \beta)^{ 1 / \bar{k}}}{1-\alpha} \Big \} \nonumber\\& = \max_{\beta \in [0,1]} \Big \{\frac{1-\alpha \vee \beta}{1-\alpha} \frac{1}{1-\alpha \vee \beta}\int_{\alpha \vee \beta }^1 \left ( \hat{F}^{-1}(u) -d \right )\mathrm{d} u + \varepsilon \cdot \frac{(1 - \alpha \vee \beta)^{ 1 / \bar{k}}}{1-\alpha} \Big \} \nonumber\\& = \frac{1}{1-\alpha}\max_{\beta \in [\alpha,1]} \big \{(1- \beta) \big ( \mathrm{ES}_{\beta} (X^{\hat{F}} )-d \big ) + \varepsilon (1 - \beta)^{ 1 / \bar{k}} \big \} .\end{align}

In particular, if we take $\alpha=0$ , then $\mathrm{ES}_0= \mathbb{E} $ . The expression (3.9) implies that

(3.10) \begin{align}\sup_{F \in \mathcal S (\hat{F};\, k, \varepsilon)} \mathbb{E} \left [ ( X^{F} -d)_+ \right ]& = \max_{\beta \in [0,1]} \big \{ (1 - \beta)\big ( \mathrm{ES}_\beta (X^{\hat{F}} )-d \big ) + \varepsilon (1-\beta)^{ 1 - 1 /k } \big \},\end{align}

which recovers the result of Proposition 2 in Guan et al. (Reference Guan, Jiao and Wang2023).

3.2. Limited loss transform

In this subsection, we solve the problem (1.4) when $\ell= \ell^m$ and $\rho$ is the distortion risk measure $ \rho^g$ , as represented in (2.2). Specifically, we solve the following optimization problem:

(3.11) \begin{align} \sup_{F \in {\mathcal S }(\hat{F};\, k, \varepsilon)} \rho(X^F \wedge m) = \sup \big \{\rho^g \left (X^F \wedge m \right)\,:\, F \in {\mathcal S} (\hat{F};\, k, \varepsilon) \big \}. \end{align}

To avoid discussing the trivial case, we assume $ m \lt \mathrm{ess\mbox{-}sup} X^{\hat{F}}.$ To solve the problem (3.11), we denote $ q_1 \triangleq \hat{F} (m) $ and

(3.12) \begin{align} q_0^k & \triangleq \inf \left\{ q \geqslant 0\,: \int_q^{q_1} \big \vert m - \hat{F}^{-1} (u) \big \vert^k \, \mathrm{d} u \leqslant \varepsilon^k \right\} \\ & = \inf \left\{ q \geqslant 0\,: \int_q^1 \big \vert m - \hat{F}^{-1} (u) \wedge m \big \vert^k \, \mathrm{d} u \leqslant \varepsilon^k \right\}\! . \nonumber\end{align}

We point out that if $ q_0^k = 0 $ , then

\begin{align*} \big (W_k ( m \vee \hat{F}^{-1}, \, \hat{F}^{-1})\big )^k = \int_0^{1} \big \vert m \vee \hat{F}^{-1} (u)- \hat{F}^{-1} (u) \big \vert^k \, \mathrm{d} u = \int_0^{q_1} \big \vert m - \hat{F}^{-1} (u) \big \vert^k \, \mathrm{d} u \leqslant \varepsilon^k ,\end{align*}

that is, $m \vee \hat{F}^{-1} \in \mathcal Q (\hat{F}^{-1}, k,\varepsilon)$ . It is easy to see that $ \rho \big ( ( m \vee \hat{F}^{-1} )\wedge m \big ) = \rho(m) = m $ . Meanwhile, $F^{-1} \wedge m \leqslant m $ for any quantile function $F^{-1}$ . By the monotonicity of $ \rho $ , we know that $ \rho ( X^F \wedge m ) \leqslant m $ for any $ F \in \mathcal S(\hat{F}^{-1}, k,\varepsilon)$ . Therefore, $ q_0^k = 0 $ implies $ \sup_{F \in {\mathcal S }(\hat{F};\, k, \varepsilon)} \rho(X^F \wedge m) =m$ , which is a trivial case in which the worst-case risk measure is equal to the limit of $m$ and it is achieved at a worst-case distribution $ m \vee \hat{F}^{-1} \in \mathcal S(\hat{F}^{-1}, k,\varepsilon) $ . If $ q_0^k \gt 0$ , then $ m \vee \hat{F}^{-1} \notin \mathcal S(\hat{F}^{-1}, k,\varepsilon) $ . In this case, since $ \hat{F}^{-1} (u) $ is finite for any $ 0 \lt u \lt 1 $ , the integral $\int_q^1 \big \vert m - \hat{F}^{-1} (u) \wedge m \big \vert^k \, \mathrm{d} u $ is continuous in $q$ . Therefore, $ q_0^k $ satisfies the equation

(3.13) \begin{align} \int_q^{q_1} \big \vert m - \hat{F}^{-1} (u) \big \vert^k \, \mathrm{d} u= \int_q^1 \big \vert m - \hat{F}^{-1} (u) \wedge m \big \vert^k \, \mathrm{d} u = \varepsilon^k .\end{align}

The problem (3.11) is more challenging compared to the problem (3.1) because $\ell^m (x) = x \wedge m $ is a concave function. Additionally, using Lemmas 3.1 and 3.2 of Cai et al. (Reference Cai, Liu and Yin2024), we have

(3.14) \begin{align} \sup_{F \in {\mathcal S }(\hat{F};\, k, \varepsilon) } \rho (X^F\wedge m) &=m+ \sup_{F \in {\mathcal S }(\hat{F};\, k, \varepsilon) } \int_0^{F(m)}\gamma(u) \left (F^{-1}(u)-m \right )\mathrm{d} u \nonumber \\ &= m+\sup_{F \in {\mathcal S }(\hat{F};\, k, \varepsilon) } \, \min_{\beta\in[0,1]} L(\beta,F^{-1}), \end{align}

where

\begin{align*} L(\beta,F^{-1})\triangleq \int_0^1 \gamma_{2,\beta} (u)\left (F^{-1}(u)-m \right ) \, \mathrm{d} u , \,\,\,\text{ and }\,\,\, \gamma_{2,\beta} \triangleq \gamma \cdot \mathbb{I}_{[0,\beta]} .\end{align*}

This means that the problem (3.11) reduces to a sup-inf problem, whereas the problem (3.1) reduces to a sup-sup problem. Additionally, note that the weight function $ \gamma_{2,\beta} $ in $ L(\beta,F^{-1})$ for the problem (3.14) is not a non-decreasing function. As a result, the distortion risk measure induced by $ \gamma_{2,\beta} $ is not coherent. Therefore, the approach used for solving the problem (3.1) cannot be applied to the problem (3.11).

In this subsection, we first solve the problem (3.11) for $k=1$ and $k=2$ . To proceed, for any $k \geqslant 1$ , we introduce two sets of quantile functions as follows:

(3.15) \begin{align}\mathcal{A}_1 & \triangleq \big\{ F^{-1}\,:\, W_k \big( F^{-1}, \hat{F}^{-1} \big) \leqslant \varepsilon, \,\, \hat{F}^{-1} \wedge m \leqslant F^{-1} \big\}, \end{align}

(3.16) \begin{align}\mathcal A_2& \triangleq \big \{ F^{-1}\,:\, W_k \big( F^{-1}, \hat{F}^{-1} \wedge m \big) \leqslant \varepsilon, \,\,\, \hat{F}^{-1} \wedge m \leqslant F^{-1}\leqslant m \big \}. \end{align}

In the following lemma, we show that the problem (3.11) reduces to an optimization problem over the set $ \mathcal{A}_1$ or $\mathcal{A}_2$ .

Lemma 3.2. Let U be a uniform random variable on (0,1). The following equations hold:

(3.17) \begin{align} \sup_{F \in {\mathcal S }(\hat{F};\, k, \varepsilon)} \rho(X^F \wedge m)=\sup_{F^{-1} \in \mathcal{A}_1 } \rho \left ( F^{-1}(U) \wedge m \right ) = \sup_{F^{-1} \in \mathcal{A}_2 } \rho \left ( F^{-1}(U) \wedge m \right )\!.\end{align}

Proof. For any quantile function $F^{-1}$ and its distribution function $F$ , we have $ X^F \buildrel \mathrm{d} \over = F^{-1} (U) $ , $ X^F \wedge m \buildrel \mathrm{d} \over = F^{-1} (U) \wedge m $ , and we can write

(3.18) \begin{align}\rho ( X^F \wedge m ) = \int_0^1 \gamma (u) \left ( F^{-1} (u) \wedge m \right ) \mathrm{d} u.\end{align}

From the proof of Proposition 3 of Liu et al. (Reference Liu, Mao, Wang and Wei2022), we have

(3.19) \begin{align} \sup_{F \in {\mathcal S }(\hat{F};\, k, \varepsilon)} \rho(X^F \wedge m) = \sup \big \{ \rho ( X^F \wedge m) \,:\, W_k ( F, \hat{F} ) \leqslant \varepsilon \text{ and } \hat{F}^{-1} \leqslant F^{-1} \big \}.\end{align}

For any $ F \in \mathcal S(\hat{F};\, k, \varepsilon)$ satisfying $ \hat{F}^{-1} \leqslant F^{-1}$ , we see that $ \hat{F}^{-1 } \wedge m \leqslant F^{-1} \wedge m \leqslant m $ , and

\begin{align*} \big \vert F^{-1} (u) \wedge m - \hat{F}^{-1} (u) \wedge m \big \vert &= \begin{cases}\big \vert F^{-1} (u) - \hat{F}^{-1} (u) \big \vert, & \hat{F}^{-1} (u) \leqslant F^{-1} (u) \leqslant m \\ \big \vert F^{-1} (u) - m \big \vert, & \hat{F}^{-1} (u) \leqslant m \leqslant F^{-1} (u) \\ 0 , & \text{otherwise} \end{cases} \\ & \leqslant \big \vert F^{-1} (u) - \hat{F}^{-1} (u) \big \vert ,\,\,\,\,\,\,\,\,\, u \in (0,1).\end{align*}

It follows that $ W_k ( F^{-1}\wedge m, \hat{F}^{-1}\wedge m ) \leqslant W_k ( F^{-1}, \hat{F}^{-1} ) \leqslant \varepsilon$ . Thus, $ F^{-1}\wedge m \in \mathcal A_2 $ . Together with (3.18) and (3.19), we have

(3.20) \begin{align} \sup_{F \in {\mathcal S }(\hat{F};\, k, \varepsilon)} \rho(X^F \wedge m)\leqslant \sup \Big \{ \int_0^1 \gamma (u) F^{-1} (u) \mathrm{d} u\,:\, F \in \mathcal A_2 \Big \} .\end{align}

For any $ F^{-1} \in \mathcal A_2 $ , we define a quantile function $ \tilde{F}^{-1} (u)$ as

\begin{align*} \tilde{F}^{-1} (u) = \max\{F^{-1} (u) , \hat{F}^{-1}(u) \} = \begin{cases} F^{-1}(u) , & \text{ for } u \leqslant q_1, \text{ that is, } \hat{F}^{-1}(u) \leqslant m, \\ \hat{F}^{-1}(u) , & \text{ for } u \gt q_1, \text{ that is, } m \lt \hat{F}^{-1}(u) .\end{cases}\end{align*}

It is easy to see that $\tilde{F}^{-1} \geqslant \hat{F}^{-1} \geqslant \hat{F}^{-1} \wedge m $ , $ \tilde{F}^{-1}\wedge m = F^{-1} \wedge m = F^{-1}$ , and

\begin{align*}\big ( W_k ( \tilde{F}^{-1} , \hat{F}^{-1}) \big )^k = \int_0^{q_1 } \big \vert \tilde{F}^{-1} (u) - \hat{F}^{-1} (u) \big \vert^k \mathrm{d} u& = \int_0^{q_1 } \big \vert F^{-1} (u) - \hat{F}^{-1} (u) \wedge m \big \vert^k \, \mathrm{d} u\\& \leqslant \int_0^{1 } \big \vert F^{-1} (u) - \hat{F}^{-1} (u) \wedge m \big \vert^k \, \mathrm{d} u\leqslant \varepsilon^k,\end{align*}

where the last inequality holds because $ F^{-1} \in \mathcal A_2$ . Therefore, $ \tilde{F}^{-1} \in \mathcal A_1 $ and $ \rho(X^{\tilde{F}} \wedge m ) \geqslant \rho(X^{F} )$ . It implies that

(3.21) \begin{align} \sup \Big \{ \int_0^1 \gamma (u) F^{-1} (u) \, \mathrm{d} u \,:\, F \in \mathcal A_2 \Big \} \leqslant \sup \Big \{ \int_0^1 \gamma (u) \big ( \tilde{F}^{-1} (u) \wedge m \big ) \, \mathrm{d} u\,:\, \, \tilde{F} \in \mathcal A_1 \Big \} .\end{align}

From (3.18) and (3.19), write $ \sup_{F \in {\mathcal S }(\hat{F};\, k, \varepsilon)} \rho(X^F \wedge m) = \sup \Big \{ \int_0^1 \gamma (u) G^{-1} (u) \mathrm{d} u , G \in \mathcal{G} \Big \} $ where

\begin{align*}\mathcal{G} \triangleq \big \{ G^{-1} = F^{-1} \wedge m \text{ for some } F \text{ satisfies }W_k ( F, \hat{F} ) \leqslant \varepsilon \text{ and } \hat{F}^{-1} \leqslant F^{-1} \big \}.\end{align*}

Arbitrarily take $ F^{-1} \in \mathcal{A}_1 $ . It is easy to see that $\hat{F}^{-1} \leqslant \max\{F^{-1} , \hat{F}^{-1}\} $ and

\begin{align*}\max\big \{ F^{-1}(u) , \hat{F}^{-1} (u) \big \} \wedge m = \max\big \{ F^{-1}(u) \wedge m , \hat{F}^{-1} (u) \wedge m \big \} = F^{-1}(u) \wedge m.\end{align*}

Meanwhile,

\begin{align*}\big \vert \max \big \{ F^{-1}(u) , \hat{F}^{-1} (u) \big \}- \hat{F}^{-1} (u) \big \vert = \big ( F^{-1}(u) - \hat{F}^{-1} (u) \big )_+ \leqslant \big \vert F^{-1}(u) - \hat{F}^{-1} (u) \big \vert\end{align*}

implies $ W_k \big (\max\big \{ F^{-1} , \hat{F}^{-1} \big \}, \hat{F}^{-1} \big ) \leqslant W_k ( F, \hat{F} ) \leqslant \varepsilon$ , where the second inequality is due to $ F^{-1} \in \mathcal{A}_1 $ . Therefore, $F^{-1} \wedge m = \max \big \{ F^{-1} , \hat{F}^{-1} \big \} \wedge m \in \mathcal{G}.$ It follows that

\begin{align*}\sup \Big \{ \int_0^1 \gamma (u) \left ( F^{-1} (u) \wedge m \right ) \mathrm{d} u\,:\, \, F^{-1} \in \mathcal A_1 \Big \} \leqslant \sup \Big \{ \int_0^1 \gamma (u) G^{-1} (u) \mathrm{d} u , G \in \mathcal{G} \Big \} = \sup_{F \in {\mathcal S }(\hat{F};\, k, \varepsilon)} \rho(X^F \wedge m) .\end{align*}

The above inequality, together with (3.20) and (3.21), imply that (3.17) is achieved as desired.

Now, we first apply Lemma 3.2 to solve the problem (3.11) when $k=1$ .

Theorem 3.3 Assume $k=1$ and $ m \lt \mathrm{ess\mbox{-}sup} X^{\hat{F}} $ in the problem (3.11). Then, the quantile function of the worst-case distribution $F^*$ for the problem (3.11) is

(3.22) \begin{align}(F^*)^{-1} (u) = \begin{cases}\hat{F}^{-1}(u) , & 0 \leqslant u \leqslant q_0^1, \\ m , & q_0^1 \lt u \leqslant q_1, \\ \hat{F}^{-1} (u) , & q_1 \lt u \leqslant 1 , \end{cases}\end{align}

where $q_1 = \hat{F}(m)$ and $ q_0^1 $ is defined in (3.12). Furthermore, the worst-case risk measure under $F^*$ is

(3.23) \begin{align} \sup_{F \in {\mathcal S }(\hat{F};\, 1, \varepsilon)} \rho(X^F \wedge m) = \min\big \{ \rho (X^{\hat{F}} ) +\varepsilon, \, m \big \} =\begin{cases} \rho (X^{\hat{F}} ) +\varepsilon , & q_0^1 \gt 0, \\ m , & q_0^1 = 0. \end{cases}\end{align}

Proof. The proof of this theorem is given in the appendix.

Remark 3.2. We make two noteworthy observations about Theorem 3.3. First, the worst-case distribution, or equivalently, the worst-case quantile function given in (3.22) depends only on the limit $m$ , the reference distribution $\hat{F}$ , and the radius $\varepsilon$ , and is independent of the coherent distortion risk measure $\rho$ . Specifically, when $k=1$ , the Wasserstein distance between two distributions ordered by the first-order stochastic dominance (FSD) simplifies to the difference of their means. Consequently, if a law-invariant risk measure $\rho$ preserves FSD and the convex order, similar arguments used in the proof of Theorem 3.3 can be applied to derive the worst-case distribution in (3.22). Thus, the result in Theorem 3.3 holds for any coherent distortion risk measure that preserves FSD and the convex order.

Second, from (3.23), we have $ \sup_{F \in {\mathcal S }(\hat{F};\, 1, \varepsilon)} \rho(X^F \wedge m) = \rho (X^{\hat{F}} ) + \varepsilon $ if $q_0^1 \gt 0 $ . According to Proposition 4 of Liu et al. (Reference Liu, Mao, Wang and Wei2022), it follows that $ \sup\{\rho (X^F) \,:\, W_1 (F,\hat{F}) \leqslant \varepsilon \} = \rho (X^{\hat{F}} ) +\varepsilon $ . Therefore, regardless of the choice of $\rho$ , when $ m \gt \rho (X^{\hat{F}} ) +\varepsilon$ , the worst-case distribution should fully utilize the distance tolerance $\varepsilon$ between the reference distribution and a candidate distribution in the left tail (before hitting $m$ ). This ensures that the difference between the worst-case risk measure value and $\rho (X^{\hat{F} } ) $ remains $ \varepsilon$ . Mathematically, this implies that $ \sup_{F \in {\mathcal S }(\hat{F};\, 1, \varepsilon)} \rho(X^F \wedge m) = \rho (X^{\hat{F}} ) + \varepsilon $ is a constant for all $m \in [ \rho (X^{\hat{F}} ) + \varepsilon, \, \mathrm{ess\mbox{-}sup} X^{\hat{F}}] $ .

Additionally, we note that (3.22) indicates that if the reference distribution $\hat{F}$ is non-negative, then the worst-case distribution $F^*$ for problem (3.11) is also non-negative. Furthermore, it serves as the worst-case distribution for the problem $\sup_{F \in {\mathcal S}^+ (\hat{F};\, 1, \varepsilon)} \rho(X^F \wedge m)$ with $\sup_{F \in {\mathcal S}^+ (\hat{F};\, 1, \varepsilon)} \rho(X^F \wedge m) =\sup_{F \in {\mathcal S} (\hat{F};\, 1, \varepsilon)} \rho(X^F \wedge m)$ , whose expression is given in (3.23).

Next, we solve the problem (3.11) when $k=2$ . To do so, we introduce the concept of isotonic projection. Let $\mathcal{K}= \big \{ k \,:\,(0,1)\mapsto\mathbb{R} \ \big \vert \ \int_0^1 k (u)^2\mathrm{d} u\lt \infty, \, k \, \text{is non-decreasing and left-continuous} \big\} $ be the space of square-integrable, non-decreasing and left-continuous functions on (0,1). Denote the isotonic projection of a function $ f \in L^2(0,1) $ onto the space $\mathcal{K} $ as $ f^\uparrow=\mathrm{arg\,min}_{k \in\mathcal{K}}|| f -k ||_2.$ The properties of isotonic projection are discussed in Rüschendorf and Vanduffel (Reference Rüschendorf and Vanduffel2020) and the references therein.

Theorem 3.4. Suppose that Assumption 2.1 holds with $k=2$ and $m \lt \mathrm{ess\mbox{-}sup} X^{\hat{F}} $ in the problem (3.11), and $q_0^2 \gt 0$ . Then, there exists a worst-case distribution $F^* \in {\mathcal S}(\hat{F};\, 2, \varepsilon)$ such that $ \sup_{F \in {\mathcal S }(\hat{F};\, 2, \varepsilon)} \rho(X^F \wedge m)= \rho \left ( X^{F^*} \wedge m \right ) $ , and

(3.24) \begin{align}(F^*)^{-1} (u) = \begin{cases} \hat{F}^{-1}(u) + \lambda^* \, \gamma(u) , & 0 \lt u \leqslant \theta^*, \\ m , & \theta^* \lt u \leqslant q_1, \\ \hat{F}^{-1}(u) , & q_1 \lt u \lt 1 , \end{cases}\end{align}

where $\lambda^* \geqslant 0 $ and $\theta^* \in (0,q_1) $ satisfy $ W_2 ( F^* , \hat{F}) = \varepsilon$ . Furthermore, the worst-case risk measure value under $F^*$ is

(3.25) \begin{align} \sup_{F \in {\mathcal S }(\hat{F};\, 2, \varepsilon)} \rho(X^F \wedge m) = \rho( X^{\hat{F}} ) + \lambda^* \int_0^{\theta^* } \gamma(u)^2 \, \mathrm{d} u + \int_{\theta^* }^{q_1} \big ( m - \hat{F}^{-1}(u) \big ) \gamma(u) \, \mathrm{d} u .\end{align}

Proof. The proof of this theorem is given in the appendix.

Remark 3.3. Unlike the worst-case quantile in (3.22) for $k=1$ , the worst-case quantile in (3.24) for $k=2$ depends not only on the uncertainty set but also the risk measure $\rho$ with the weight function $\gamma$ . However, similar to the case when $k=1$ , (3.24) implies that if the reference distribution $\hat{F}$ is non-negative, then the worst-case distribution $F^*$ for the problem (3.11) with $k=2$ is also non-negative. Furthermore, $F^*$ serves as the worst-case distribution for the problem $\sup_{F \in {\mathcal S}^+ (\hat{F};\, 2, \varepsilon)} \rho(X^F \wedge m)$ , with $\sup_{F \in {\mathcal S}^+ (\hat{F};\, 2, \varepsilon)} \rho(X^F \wedge m) =\sup_{F \in {\mathcal S} (\hat{F};\, 2, \varepsilon)} \rho(X^F \wedge m)$ , whose expression is given in (3.25).

Additionally, we note that the worst-case quantile functions for $k=1$ and $k=2$ share a common feature in the right tail: there exists $0 \lt p \leqslant q_1$ such that $ (F^*)^{-1}(u) = m$ for $ p \lt u \leqslant q_1$ and $(F^*)^{-1}(u) = \hat{F}^{-1}(u) $ for $ q_1 \lt u \lt 1$ . Indeed, we can show that this feature in the right tail of a worst-case distribution holds true for all $k \geqslant 1$ , as presented in the following proposition.

Remark 3.4. For $k=2$ , following the argument used in Theorem 4.8 of Bernard et al. (Reference Bernard, Pesenti and Vanduffel2024), one has

\begin{align*}\sup_{F \in {\mathcal S}( \hat{F};\, 2, \varepsilon) } \rho (\ell (X^F) ) = \sup_{ (\hat{\mu} - \mu)^2 + (\hat{ \sigma} - \sigma)^2 \leqslant \varepsilon^2 \atop \sigma \gt 0} \sup_{F \in {\mathcal S}( \hat{F};\, 2, \varepsilon, \mu,\sigma) } \rho (\ell (X^F) ) .\end{align*}

The inner problem on the right hand of the equation, that is, $ \sup_{F \in {\mathcal S}( \hat{F};\, 2, \varepsilon, \mu,\sigma) } \rho (\ell (X^F) ) $ , is studied in Cai et al. (Reference Cai, Liu and Yin2024) for stop-loss and limited loss $\ell$ , and its optimal solutions have a relatively complex mathematical structure. Therefore, the two-step optimization problem is not mathematically tractable.

Proposition 3.5. Suppose that Assumption 2.1 holds with $k \geqslant 1$ and $m \lt \mathrm{ess\mbox{-}sup} X^{\hat{F}} $ in the problem (3.11).

If the problem $\sup \left \{ \rho(F^{-1} (U)) \,:\, F^{-1} \in \mathcal A_2 \right \} $ has a maximizer $(\tilde{F}^*)^{-1}$ , that is, $ (\tilde{F}^*)^{-1} \in \mathrm{arg\,max} \left \{ \rho(F^{-1} (U)) \,:\, F^{-1} \in \mathcal A_2 \right \},$ then the quantile $(F^*)^{-1}$ defined as

(3.26) \begin{align} (F^*)^{-1} (u) = \max\{(\tilde{F}^*)^{-1} (u), \, \hat{F}^{-1}(u) \} = \begin{cases} (\tilde{F}^*)^{-1}(u) , & 0 \leqslant u \leqslant \tilde{F}^*(m{-}), \\ m ,& \tilde{F}^*(m{-}) \lt u \leqslant q_1, \\ \hat{F}^{-1}(u), & q_1 \lt u \leqslant 1, \end{cases}\end{align}

is the worst-case quantile for the problem (3.11).

Proof. Define $(F^*)^{-1} (u) = \max\{(\tilde{F}^*)^{-1} (u) , \hat{F}^{-1}(u) \} $ . Note that $ \hat{F}^{-1} (u) \leqslant (\tilde{F}^*)^{-1} (u) \lt m $ for $ u \lt \tilde{F}^*(m{-}) $ and $(\tilde{F}^*)^{-1} (u) = m $ for $ u\gt \tilde{F}^*(m{-}) $ . Then we can further write $(F^*)^{-1}$ in (3.26). It is easy to check that $ W_k ((F^*)^{-1}, \hat{F}^{-1} ) =W_k ( (\tilde{F}^*)^{-1} , \hat{F}^{-1} \wedge m ) \leqslant \varepsilon $ and

\begin{align*} \rho ( X^{F^* } \wedge m ) & = \int_0^1 \gamma (u) \left ( (F^*)^{-1} (u) \wedge m \right ) \mathrm{d} u = \int_0^1 \gamma (u) (\tilde{F}^*)^{-1} (u) \mathrm{d} u \\ & = \sup \Big \{ \int_0^1 \gamma (u) F^{-1} (u) \mathrm{d} u \,:\, F \in \mathcal A_2 \Big \} =\sup \big \{ \rho (F^{-1}(U)) \,:\, F \in \mathcal A_2 \big \} .\end{align*}

By Lemma 3.2, $F^* $ is a worst-case distribution for the problem (3.11).

Remark 3.5. Proposition 3.5 also means that like the cases when $k=1$ and $k=2$ , if the reference distribution $\hat{F}$ is non-negative and the problem $\sup \left \{ \rho(F^{-1} (U)) \,:\, F^{-1} \in \mathcal A_2 \right \} = \sup_{ F^{-1} \in \mathcal A_2 }\rho(F^{-1} (U))$ has a maximixer $(\tilde{F}^*)^{-1} $ , then the worst-case distribution $F^*$ for the problem (3.11) with $k \geqslant 3$ is nonnegative and it is also the worst-case distribution for the problem $\sup_{F \in {\mathcal S}^+ (\hat{F};\, k, \varepsilon)} \rho(X^F \wedge m)$ with $\sup_{F \in {\mathcal S}^+ (\hat{F};\, k, \varepsilon)} \rho(X^F \wedge m) =\sup_{F \in {\mathcal S} (\hat{F};\, k, \varepsilon)} \rho(X^F \wedge m)$ . However, it is difficult to solve problem (3.11) for $k \geqslant 3$ or the problem $\sup \left \{ \rho(F^{-1} (U)) \,:\, F^{-1} \in \mathcal A_2 \right \} = \sup_{ F^{-1} \in \mathcal A_2 }\rho(F^{-1} (U))$ for $k \geqslant 3$ in Proposition 3.5. We expect that new approaches can be developed for solving problem (3.11) when $k \geqslant 3$ .

3.3. Limited stop-loss transform

In this subsection, we solve the problem (1.4) when $\ell= \ell_d^m$ and $\rho$ is the distortion risk measure $ \rho^g$ , as represented in (2.2). Specifically, we solve the following optimization problem:

(3.27) \begin{align}\sup_{F \in {\mathcal S } (\hat{F};\, k, \varepsilon)} \rho \left ((X^F-d)_+ \wedge m \right ) = \sup\big \{\rho^g \left ( ( X^F-d)_+ \wedge m \right)\,:\, F \in {\mathcal S} (\hat{F};\, k, \varepsilon)\big \}.\end{align}

To avoid discussing trivial cases, for the problem (3.27), we assume $ \mathrm{ess\mbox{-}inf} (X^{\hat{F}} ) \lt d \lt m+d \lt \mathrm{ess\mbox{-}sup} (X^{\hat{F}}).$ Note that the stop-loss function $ (x-d)_+$ is convex in $x$ , and the limited loss function $x \wedge m $ is concave in $x$ , but, the limited loss function $ (x-d)_+ \wedge m $ is neither convex nor concave in $x$ . Hence, we cannot apply the arguments used in Sections 3.1 and 3.2 to solve problem (3.27). However, we can reformulate problem (3.27) so that we can use the solutions in Section 3.2 to solve the problem (3.27). In Section 3.2, solutions for problem (3.11) are available only when $k=1$ and $k=2$ . Thus, for the problem (3.27), we only consider the cases when $k=1$ and $k=2$ .

Recall the set of quantile functions $\mathcal Q_d (\hat{F}^{-1};\, k, \varepsilon)= \{\hat{G}^{-1}\,:\, \hat{G}^{-1}+d \in \mathcal Q ( \hat{F}^{-1};\, k, \varepsilon) \}$ defined in the proof of Theorem 3.1 and $\rho_{1,\beta_1}$ is a coherent distortion risk measure induced by distortion function $g_{1,\beta_1}$ defined in (A1). Thus, we reduce the problem (3.27) to a problem involving the problem (3.11), as stated in the following proposition:

Proposition 3.6. Suppose that Assumption 2.1 holds, $ m \gt 0$ , and $ \mathrm{ess\mbox{-}inf} (X^{\hat{F}} ) \lt d \lt d+m \lt \mathrm{ess\mbox{-}sup} (X^{\hat{F}}) $ . Then, for $k=1,2$ , we have

(3.28) \begin{align} \sup_{F \in {\mathcal S }(\hat{F};\, k, \varepsilon)} \rho((X^F-d)_+ \wedge m) & = \sup_{0\leqslant \beta \leqslant 1} \big \{ \Vert \gamma_{1, \beta} \Vert_1 \cdot \max_{F^{-1} \in \mathcal Q_d (\hat{F}^{-1};\, k, \varepsilon) } \left \{ \rho_{1,\beta} ( X^{F} \wedge m) \right \} \big \}.\end{align}

Proof. For any $ F \in \mathcal S (\hat{F};\, k, \varepsilon)$ , we have

\begin{align*} \rho ( (X^F-d)_+ \wedge m) & = \int_{F(d) }^{F(d+m)} \gamma (u)\left ( F^{-1} (u) -d \right ) \mathrm{d} u + m \int_{F(d+m) }^1 \gamma (u) \mathrm{d} u \nonumber \\ & = \max_{0\leqslant \beta \leqslant F(d+m) } \Big \{ \int_{\beta }^{F(d+m)} \gamma (u)\big ( F^{-1} (u) -d \big ) \mathrm{d} u \Big \}+ m \Big ( 1 - \int_0^{F(d+m)} \gamma(u) \mathrm{d} u \Big ) \nonumber \\ & = \max_{0\leqslant \beta \leqslant F(d+m) } \Big \{\int_{\beta}^{F(d+m) } \gamma (u) \big ( F^{-1} (u) -d - m \big ) \mathrm{d} u + m- m \int_0^{\beta} \gamma(u) \mathrm{d} u \Big \}\\ & = \max_{0\leqslant \beta \leqslant F(d+m) } \Big \{ \int_0^{F(d+m) } \gamma_{1, \beta} (u) \left ( F^{-1} (u) -d - m \right ) \mathrm{d} u + m- m \int_0^{\beta} \gamma(u) \mathrm{d} u \Big \} , \end{align*}

where $\gamma_{1,\beta} = \gamma\cdot \mathbb{I}_{[\beta , 1]}$ with $\Vert \gamma_{1, \beta} \Vert_1 = \int_{\beta}^1 \gamma(u) \mathrm{d} u = 1 - \int_0^{\beta} \gamma(u) \mathrm{d} u$ . Using the same argument in Section 3.1, for any $\beta \lt 1 $ , we can define a coherent distortion risk measure $\rho_{1,\beta}$ induced by distortion function $g_{1,\beta}$ defined in (A1). Denote the quantile function $F^{-1}(u) - d$ by $F_{-d}^{-1}(u)$ and denote $ F(d+m)$ by $ F_{-d} (m) $ , that is, $F_{-d}^{-1} = F^{-1} - d$ and $ F(d+m) = F_{-d} (m)$ . It is easy to check that

(3.29) \begin{align} \rho ((X^F-d)_+ \wedge m ) & = \max_{0\leqslant \beta \leqslant F_{-d}(m) } \Big \{ \int_0^{F_{-d}(m) } \gamma_{1, \beta} (u) \left ( F_{-d}^{-1} (u) - m \right ) \mathrm{d} u + m- m \int_0^{\beta} \gamma(u) \mathrm{d} u \Big \} \nonumber \\ & = \max_{0\leqslant \beta \leqslant F_{-d}(m) } \Big \{ \Vert \gamma_{1, \beta} \Vert_1 \int_0^{F_{-d}(m) } \frac{\gamma_{1, \beta} (u) }{\Vert \gamma_{1, \beta} \Vert_1}\left ( F_{-d}^{-1} (u) - m \right ) \mathrm{d} u + m \Vert \gamma_{1, \beta} \Vert_1\Big \} \nonumber \\ & = \max_{0\leqslant \beta \leqslant F_{-d}(m) }\left \{ \Vert \gamma_{1, \beta} \Vert_1 \cdot \rho_{1,\beta} ( X^{F_{-d}} \wedge m) \right \}, \end{align}

where the last equality is from (A6). If $ \beta \geqslant F_{-d} (m) $ , then $ \gamma_{1, \beta } (u) = \gamma (u) \cdot \mathbb{I}_{[\beta,1]} (u) = 0 $ for all $ u \lt F_{-d} (m) $ and $\int_0^{F_{-d}(m) } \gamma_{1, \beta} (u) \left ( F_{-d}^{-1} (u) - m \right ) \mathrm{d} u = 0$ . Therefore, for any $\beta \geqslant F_{-d} (m)$ , we have

\begin{align*} \Vert \gamma_{1, \beta} \Vert_1 \cdot \rho_{1,\beta} ( X^{F_{-d}} \wedge m) \ = \ m\int_{\beta}^1 \gamma(u) \mathrm{d} u \ \leqslant \ m\int_{F(d+m)}^1 \gamma(u) \mathrm{d} u \ = \ \Vert \gamma_{1, F_{-d} (m)} \Vert_1 \cdot \rho_{1,F_{-d} (m)} ( X^{F_{-d}} \wedge m) . \end{align*}

It says that $ F_{-d} (m)$ is sub-optimal to all larger probability levels for the maximization problem (3.29). As a consequence, we have

\begin{align*} \sup_{F \in \mathcal S(\hat{F};\, k, \varepsilon) } \rho \left ((X^F-d)_+ \wedge m) \right ) & = \sup_{F \in \mathcal S(\hat{F};\, k, \varepsilon) } \Big \{ \max_{0\leqslant \beta \leqslant F_{-d}(m) } \Vert \gamma_{1, \beta} \Vert_1 \cdot \rho_{1,\beta} ( X^{F_{-d}} \wedge m) \Big \} \\ & = \sup_{F \in \mathcal S(\hat{F};\, k, \varepsilon) } \Big \{ \max_{0\leqslant \beta \leqslant 1 } \Vert \gamma_{1, \beta} \Vert_1 \cdot \rho_{1,\beta} ( X^{F_{-d}} \wedge m) \Big \} \\ & = \sup_{F^{-1} \in \mathcal Q_d (\hat{F}^{-1};\, k, \varepsilon)} \Big \{ \max_{0\leqslant \beta \leqslant 1 } \Vert \gamma_{1, \beta} \Vert_1 \cdot \rho_{1,\beta} ( X^{F} \wedge m) \Big \} \\ & = \sup_{0\leqslant \beta \leqslant 1} \Big \{ \Vert \gamma_{1, \beta} \Vert_1 \sup_{F^{-1} \in \mathcal Q_d (\hat{F}^{-1};\, k, \varepsilon)} \rho_{1,\beta} ( X^{F} \wedge m) \Big \} .\end{align*}

For any $ \beta \in [0,1]$ , the function $\gamma_{1,\beta} = \gamma \cdot \mathbb{I}_{[\beta,1]}$ is increasing and thus the risk measure $\rho_{1, \beta }$ is coherent. The problem $ \sup_{F^{-1} \in \mathcal Q_d } \rho_{1,\beta} ( X^{F} \wedge m) $ can be solved by Theorem 3.3 and Theorem 3.4 when $k=1$ and $k=2$ , respectively. Therefore, the expression (3.28) is obtained.

Using the solutions for (3.11) derived in Section 3.2, for $k=1,2$ , we can solve the problem $\max_{F^{-1} \in \mathcal Q_d (\hat{F}^{-1};\, k, \varepsilon) } \big \{ \rho_{1,\beta} ( X^{F} \wedge m) \big \} $ in (3.28) and reduce (3.11) to a feasible one-variable maximization problem.

Corollary 3.7. Suppose that Assumption 2.1 holds, $ m \gt 0$ , and $ \mathrm{ess\mbox{-}inf} (X^{\hat{F}} ) \lt d \lt d+m \lt \mathrm{ess\mbox{-}sup} (X^{\hat{F}}) $ .

1. (1) If $k=1$ , then

   (3.30) \begin{align} \sup_{F \in \mathcal S(\hat{F};\, 1, \varepsilon) } \rho \left ( (X^F-d)_+ \wedge m ) \right ) & = \max_{0\leqslant \beta \leqslant 1} \big \{ \Vert \gamma_{1, \beta} \Vert_1 \cdot \rho_{1,\beta} ( X^{F^*} \wedge m) \big \},\end{align}
   where, for any $\beta \in [0, 1]$ ,
   \begin{align*}(F^*)^{-1} (u) = \begin{cases}\hat{F}^{-1}(u)-d , & 0 \lt u \leqslant \tilde{q}_0^1, \\ m-d , & \tilde{q}_0^1 \lt u \leqslant \tilde{q}_1,\\\hat{F}^{-1} (u) -d, & \tilde{q}_1 \lt u \lt 1, \end{cases}\end{align*}
   with $ \tilde{q}_1 = \hat{F} (m+d )$ , $ \tilde{q}_0^1 = \inf \big \{q \geqslant 0 \,:\, \int_q^{q_1} \big \vert m - \hat{F}^{-1} (u) + d \big \vert \, \mathrm{d} u \leqslant \varepsilon \big \}$ .

2. (2) If $k=2$ , then

   (3.31) \begin{align} \sup_{F \in \mathcal S(\hat{F};\, 2, \varepsilon) } \rho \left ( (X^F-d)_+ \wedge m ) \right ) & = \max_{0\leqslant \beta \leqslant 1} \big \{ \Vert \gamma_{1, \beta} \Vert_1 \cdot \rho_{1,\beta} ( X^{F^*_{\beta}} \wedge m) \big \},\end{align}
   where, for any $\beta \in [0, 1]$ ,
   \begin{align*}(F^{*}_{\beta})^{-1} (u) = \begin{cases} \hat{F}^{-1}(u) + \lambda^* \, \gamma_{1,\beta_1}(u) -d , & 0 \lt u \leqslant \theta^* , \\ m-d , & \theta^* \lt u \leqslant \tilde{q}_1 , \\ \hat{F}^{-1}(u)-d , & \tilde{q}_1 \lt u \lt 1, \end{cases}\end{align*}
   with $\lambda^* \gt 0 $ and $\theta^* \in (0,1) $ satisfy $ W_2 ( F^*_{\beta} , \hat{F}-d) = \varepsilon$ .

Proof. For any $\beta \in [0, 1]$ , (3.22) implies that $(F^*)^{-1}$ is the maximizer of $ \max_{F^{-1} \in \mathcal Q_d } \left \{ \rho_{1,\beta} ( X^{F} \wedge m ) \right \}$ when $k=1$ , while (3.24) implies that $ (F^{*}_{\beta_1})^{-1}$ is the maximizer of $ \max_{F^{-1} \in \mathcal Q_d } \left \{ \rho_{1,\beta} ( X^{F} \wedge m) \right \}$ when $k=2$ . Then the statement of the corollary follows consequently.

Both (3.30) and (3.31) are feasible one-variable maximization problems and can be easily solved numerically. We will illustrate their applications in the next section.

4. Numerical illustrations of worst-case distributions, worst-case risk measures, and their applications

In a reinsurance treaty, an insurer cedes part of its underlying insurance loss $X$ to a reinsurer. The ceded loss, denoted by $\ell(X)$ , is determined by the ceded loss function $\ell(x)$ . The reinsurer charges a (reinsurance) premium based on a premium principle $\pi$ , so the premium paid by the insurer is $\pi(\ell(X))$ , where $\pi$ is a mapping from random variables to real numbers. Specifically, $\pi(\ell(X))= (1+\theta) \mathbb{E}[ \ell(X) ]$ is known as the expected value principle, while $\pi(\ell(X))= (1+\theta) \rho(\ell(X))$ is called the (extended) distortion premium principle, where $\theta \geqslant 0$ and $\rho$ is a distortion risk measure. Additionally, if $\ell(X)=(X-d)_+$ , the (net) premium $\mathbb{E}[(X-d)_+] $ is referred to as the stop-loss premium. If $\ell(X) = X \wedge m$ , the (net) premium $\mathbb{E}[X \wedge m] $ is known as the limited expected value, representing the net premium of an insurance or reinsurance policy with a payment limit of $m$ . For applications of the transforms of $(x-d)_+$ , $x \wedge m$ , and $(x-d)_+ \wedge m$ in insurance, we refer to Albrecher et al. (Reference Albrecher, Beirlant and Teugels2017), Klugman et al. (Reference Klugman, Panjer and Willmot2019), and references therein.

In general, calculating the premium $\pi(\ell(X))$ requires complete information about the distribution of the underlying insurance loss $X$ . However, when the distribution of $X$ is uncertain, the premium $\pi(\ell(X))$ is often unknown. In such cases, the reinsurer needs to estimate the premium $\pi(\ell(X))$ . Suppose that the reinsurer uses the uncertainty set $ {\mathcal S }$ to represent all possible distributions of $X$ and has an initial estimate $\hat{F}$ for the distribution of $X$ . Based on this estimate, the reinsurer might determine the premium as $\pi(\ell(X^{\hat{F}}))$ . However, this approach may yield a premium that is lower than necessary, as it does not account for the distribution uncertainty.

In fact, when the distribution of $X$ is uncertain, the reinsurer is more interested in understanding the highest premium that could be potentially be required, given this uncertainty. Therefore, the reinsurer is concerned with the problem of finding $\sup_{F \in {\mathcal S}}\pi(\ell(X^F))$ , which represents the sharp upper bound of the premium that accounts for all possible distributions within the uncertainty set ${\mathcal S }$ .

If ${\mathcal S}$ is the set of distributions with given mean and variance, or with given moments up to a higher order, the upper bounds for the stop-loss premium $\mathbb{E}[(X-d)_+] $ have been well studied in the literature (see, for instance, Jansen et al., Reference Jansen, Haezendonck and Goovaerts1986; Kaas and Goovaerts, Reference Kaas and Goovaerts1986, and the references therein). When ${\mathcal S}$ consists of distributions lying within a Wasserstein ball and satisfying constraints on the first two moments, and $\rho$ is a coherent distortion risk measure, sharp upper bounds for the distortion premiums associated with stop-loss reinsurance $(X-d)_+ $ and limited loss reinsurance $X \wedge m$ have been derived in Cai et al. (Reference Cai, Liu and Yin2024). Utilizing the results from Section 3 of this paper, we can extend this analysis to obtain sharp upper bounds for distortion premiums associated with limited stop-loss reinsurance. Specifically, when the distribution of $X$ is uncertain and belongs to $ \mathcal S(\hat{F};\, k, \varepsilon)$ , and $\rho$ is a coherent distortion risk measure, we are able to provide sharp upper bounds for the distortion premiums associated with stop-loss reinsurance $(X-d)_+$ , limited loss reinsurance $X \wedge m$ and limited stop-loss reinsurance $(X-d)_+ \wedge m$ .

To illustrate the results derived in Section 3 and their applications, in this section, we employ Wang’s premium principle along with numerical examples. This approach will help us analyze the effects of the reference distribution $\hat{F}$ , the radius $ \varepsilon $ in the Wasserstein ball $ \mathcal S(\hat{F};\, k, \varepsilon)$ , and retention levels on the premium of limited stop-loss reinsurance.

Wang’s premium principle, introduced in Wang (Reference Wang2000), is a special distortion premium principle that utilizes a specific distortion risk measure $\rho_\alpha$ that has the following distortion function:

(4.1) \begin{equation}g_\alpha(u)=\Phi(\Phi^{-1}(u)+ \alpha ), \,\,\,0 \lt u \lt 1, \,\,\,\,\,\, \alpha \in (0,1),\end{equation}

where $\Phi^{-1}(u)$ is the quantile function of the standard normal distribution function $\Phi(x)$ . The distortion function $g_\alpha(u)$ has a weight function $\gamma_\alpha(u)=\partial_-g(x)|_{x=1-u} =g'(1-u)= e^{-\alpha \Phi^{-1}(1-u) - \frac{\alpha^2}{2}}$ , which is non-negative and increasing in $u \in (0, 1)$ for any $\alpha \in (0, 1)$ . As illustrated in Wang (Reference Wang2000), the distortion function or the transform defined in (4.1) has various interesting applications in pricing insurance and finance products.

In this section, we assume $k=2$ in the Wasserstein ball $ \mathcal S(\hat{F};\, k, \varepsilon)$ . We point out that the weight function $\gamma_\alpha(u) = g'_\alpha(u) = e^{-\alpha \Phi^{-1}(u) - \frac{\alpha^2}{2}}$ satisfies Assumption 2.1. Specifically, for any $\alpha \in (0,1) $

\begin{align*} \Vert \gamma_\alpha \Vert_2^2 &= \int_0^1 \gamma^2 (u) \mathrm{d} u = e^{-\alpha^2} \int_0^1 e^{-2\alpha \Phi^{-1}(u) } \mathrm{d} u = e^{-\alpha^2} \int_{-\infty}^{+\infty } e^{-2\alpha x}\, \mathrm{d} \Phi(x) = e^{-\alpha^2} \mathbb{E} [e^{-2\alpha Z }] = e^{-\alpha^2} e^{\frac{4\alpha^2}{2}} \lt \infty, \end{align*}

where $Z$ follows the standard normal distribution $ \Phi $ and $ \mathbb{E} [e^{-2\alpha Z }] =e^{\frac{4\alpha^2}{2}}$ is the moment generating function of $Z$ .

In the first example of this section, we illustrate the worst-case distributions for the problems of $\sup_{F \in \mathcal S(\hat{F};\, k, \varepsilon)} \rho_\alpha((X^F -d)_+)$ , $\sup_{F \in \mathcal S(\hat{F};\, k, \varepsilon)} \rho_\alpha(X^F \wedge m)$ , and $\sup_{F \in \mathcal S(\hat{F};\, k, \varepsilon)} \rho_\alpha((X^F-d)_+ \wedge m)$ .

Example 4.1. Assume that the reference distribution $\hat{F}$ in the set $ \mathcal S(\hat{F};\, k, \varepsilon)$ is a Pareto distribution given by $\hat{F}(x)=1-(\frac{12}{x+12})^4$ for $x\geqslant 0$ , with $\varepsilon=2$ and $k=2$ . Using Theorem 3.1(iii), Theorem 3.4, and Proposition 3.6, we can identify the quantile functions $(F^*)^{-1}$ of the worst-case distributions $F^*$ such that $\sup_{F \in \mathcal S(\hat{F};\, k, \varepsilon)} \rho_\alpha((X^F -d)_+) =\rho_\alpha((X^{F^*} -d)_+)$ , $\sup_{F \in \mathcal S(\hat{F};\, k, \varepsilon)} \rho_\alpha(X^{F} \wedge m)=\rho_\alpha(X^{F^*} \wedge m) $ , and $\sup_{F \in \mathcal S(\hat{F};\, k, \varepsilon)} \rho_\alpha((X^{F} -d)_+ \wedge m)=\rho_\alpha((X^{F^*}-d)_+ \wedge m)$ . The quantile functions $(F^*)^{-1}$ , with $\alpha =0.5$ for different deductibles d and limits m, are plotted in Figures 1–2, respectively.

Figure 1.

The quantile functions of the worst-case distributions for stop-loss ( $d=5,10,15 $ and $M = \infty$ ) and limited loss ( $d=0$ and $M=5,10,15$ ) reinsurance.

Figure 2.

The quantile function of the worst-case distributions for limited stop-loss reinsurance.

From these figures, we observe that the worst-case quantile function dominates the quantile function of the reference distribution in different ways. Specifically, for the stop-loss reinsurance, the worst-case quantile function exceeds the quantile function of the reference distribution in the right tails. In contrast, for the limited loss reinsurance, the worst-case quantile function dominates in the left tails. However, for the limited stop-loss reinsurance, the worst-case quantile function dominates in the middle range. These numerical results are consistent with the objectives of these reinsurances. In fact, for stop-loss reinsurance and limited loss reinsurance, the left-tail and right-tail risks of the reinsurer are eliminated, respectively. In the case of limited stop-loss reinsurance, both left-tail and right-tail risks of the reinsurer are eliminated. Therefore, the worst-case scenarios for the premiums of stop-loss reinsurance, limited loss reinsurance, and limited stop-loss reinsurance are expected to primarily involve right-tail, left-tail, and middle-range risks, respectively.

In the following example, we illustrate the effects of the radius $\varepsilon$ and the reference distribution $\hat{F}$ in $\mathcal S(\hat{F};\, k, \varepsilon)$ , as well as the retention levels $d$ and $m$ on Wang’s premium for limited stop-loss reinsurance $(X-d)_+ \wedge m$ . Specifically, we examine how these factors influence $\sup_{F \in \mathcal S(\hat{F};\, k, \varepsilon)} \rho_\alpha((X^{F} -d)_+ \wedge m)$ , which represents the highest premium that could be expected when the distribution of the underlying insurance loss $X$ is uncertain and belongs to the set $ \mathcal S(\hat{F};\, k, \varepsilon)$ .

We consider a limited stop-loss reinsurance with the ceded loss function $\ell_d^m(x) = (x - d)_+ \wedge m$ . To illustrate the effect of those factors on $\sup_{F \in \mathcal S(\hat{F};\, 2, \varepsilon)} \rho_\alpha((X^{F} -d)_+ \wedge m)$ , we calculate $\sup_{F \in \mathcal S(\hat{F};\, 2, \varepsilon)} \rho_\alpha((X^{F} -d)_+ \wedge m)$ when $\hat{F}=\hat{F}_1$ and $\hat{F}=\hat{F}_2$ with $\alpha=0.5$ for different values of $\varepsilon$ , $d$ , and $m$ .

In Figure 3 and columns 1–3 in Table 1, we examine how the radius $\varepsilon $ of the Wasserstein ball affects $\sup_{F \in \mathcal S(\hat{F};\, 2, \varepsilon)} \rho_{0.5}((X^{F} -d)_+ \wedge m)$ . To do this, we vary $\varepsilon $ from 0.1 to 1.9, while keeping $d=5$ and $m=5$ constant. In Table 1, the values in the column under the heading $\text{EXP}^{w}$ represent $\sup_{F \in \mathcal S(\hat{F}_1;\, 2, \varepsilon)} \rho_{0.5}((X^{F} -d)_+ \wedge m)$ with the exponential reference distribution $\hat{F}_1$ . The values in the column under the heading $\text{PAR}^{w}$ represent $\sup_{F \in \mathcal S(\hat{F}_2;\, 2, \varepsilon)} \rho_{0.5}((X^{F} -d)_+ \wedge m)$ with the Pareto reference distribution $\hat{F}_2$ . Additionally, the values in the column under the heading $\text{EXP}^{r}$ represent $ \rho_{0.5}((X^{{F}_1} -d)_+ \wedge m)$ with the exponential reference distribution $\hat{F}_1$ . The values in the column under the heading $\text{PAR}^{r}$ represent $\rho_{0.5}((X^{{F}_2} -d)_+ \wedge m)$ with the Pareto reference distribution $\hat{F}_2$ . From these calculations, we observe that as $\varepsilon$ increases – indicating a larger uncertainty set for the distribution of $X$ – the worst-case Wang’s premium also rises. In other words, greater uncertainty about the underlying loss distribution necessitates a higher premium to cover the worst-case scenarios. This reflects the intuitive notion that higher uncertainty about the distribution of losses requires a higher premium to safeguard against potential worst-case scenarios.

Figure 3.

${\small \sup_{F \in \mathcal S(\hat{F}; 2, \varepsilon)} \rho_{0.5}((X^{{F}} -5)_+ \wedge 5)} \text{vs} \varepsilon$ .

Table 1.

Wang’s premiums for the limited stop-loss reinsurance $(X-d)_+ \wedge m$ .

In Figure 4 and columns 4–8 in Table 1, we analyze how the limit $m$ affects $\sup_{F \in \mathcal S(\hat{F};\, 2, \varepsilon)} \rho_{0.5}((X^{F} -d)_+ \wedge m)$ . We vary $m$ from 4 to 13, while keeping $d=5$ and $\varepsilon =2 $ fixed. Additionally, in Figure 5 and columns 9–13 in Table 1, we examine how the deductible $d$ affects $\sup_{F \in \mathcal S(\hat{F};\, 2, \varepsilon)} \rho_{0.5}((X^{F} -d)_+ \wedge m)$ . We vary $d$ from 0.5 to 9.5, while keeping $m=5$ and $\varepsilon =2 $ fixed. From these calculations, we observe that as $m$ increases – indicating a larger amount of loss covered by the reinsurer – the worst-case Wang’s premium also increases. Conversely, as $d$ increases – indicating that the reinsurer covers less of the underlying insurance loss – the worst-case Wang’s premium decreases.

Figure 4.

$ {\small \sup_{F \in \mathcal S(\hat{F};\, 2, 2)} \rho_{0.5}((X^{{F}} -5)_+ \wedge m) } \, \text{vs} \, m$ .

Figure 5.

${\small \sup_{F \in \mathcal S(\hat{F};\, 2, 2)} \rho_{0.5}((X^{{F}} -d)_+ \wedge 5)} \, \text{vs} \, d$ .

Furthermore, form Figures 3–5 and Table 1, we also observe that Wang’s premiums for the limited stop-loss reinsurance with the exponential reference distribution is higher than that with the Pareto reference distribution. For instance, $\rho_{0.5}((X^{\hat{F}_1}-5)_+\wedge 5)=1.5535$ is greater than $\rho_{0.5}((X^{\hat{F}_2}-5)_+\wedge 5)=1.4748$ . These results are also reasonable. In fact, we have $\mathbb{E}[(X^{\hat{F}_1}-5)_+\wedge 5] = 0.817679 $ , which is greater than $\mathbb{E}[(X^{\hat{F}_2}-5)_+\wedge 5] =0.757744$ , and $\text{var}((X^{\hat{F}_1}-5)_+\wedge 5) =2.58943 $ , which is greater than $\text{var}((X^{\hat{F}_2}-5)_+\wedge 5) = 2.57043.$ Thus, in these senses, the exponential distribution is risker than the Pareto distribution for the limited stop-loss reinsurance $(X-5)_+ \wedge 5$ . These numerical results along with the fact that the reinsurer, selling the limited stop-loss reinsurance $(X-5)_+ \wedge 5$ , is mainly concerned with the middle-range risks. The exponential survival function dominates in left tail, while the Pareto survival function dominates in right tail. Indeed, for the two distributions $\hat{F}_1$ and $\hat{F}_2$ , it holds that $\Pr ( X^{F_1} \gt x)\gt \Pr(X^{F_2} \gt x)$ for $0 \lt x \lt 8.80321$ , while $\Pr( X^{F_1} \gt x ) \lt \Pr( X^{F_2} \gt x)$ for $x \gt 8.80321$ . In these senses, the exponential distribution is risker than the Pareto distributions for the limited stop-loss reinsurance $(X-5)_+ \wedge 5$ .

However, if we consider the stop-loss reinsurance $(X-5)_+$ , we have $\mathbb{E}[(X^{\hat{F}_1}-5)_+] = 9.16815$ , which is less than $ \mathbb{E}[(X^{\hat{F}_2}-5)_+] = 23.917$ , and $\text{var}((X^{\hat{F}_1}-5)_+) =7.85479 $ , which is less than $\text{var}((X^{\hat{F}_2}-5)_+) =21.9376.$ Thus, in these senses, the Pareto distribution is risker than the exponential distribution for the stop-loss distribution $(X-5)_+$ . Using the results derived in Section 3, we can also calculate Wang’s premiums $\rho_{0.5}((X^{\hat{F}_1}-5)_+)$ , $\rho_{0.5}((X^{\hat{F}_2}-5)_+)$ , $\sup_{F \in \mathcal S(\hat{F}_1;\, 2, \varepsilon)} \rho_{0.5}((X^{{F}} -d)_+)$ and $\sup_{F \in \mathcal S(\hat{F}_2;\, 2, \varepsilon)} \rho_{0.5}((X^{{F}} -d)_+)$ for the stop-loss reinsurance $(X-5)_+$ and conclude that the Pareto distribution is risker than the exponential distributions for the stop-loss reinsurance $(X-5)_+$ . Detailed calculations for the stop-loss reinsurance $(X-5)_+$ are omitted.

In addition, form Figures 3–5 and Table 1, we also observe that the worst-case Wang’s premiums or the highest Wang’s premiums for the limited stop-loss reinsurance is significantly higher than those calculated based on the reference distributions. For instance,

\begin{align*} \sup_{F \in \mathcal S(\hat{F}_1;\, 2, 1.9)} \rho_{0.5}((X^{{F}} -5)_+ \wedge 5) = 2.7231 \gt 1.5535 = \rho_{0.5}((X^{\hat{F}_1}-5)_+ \wedge 5) = \sup_{F \in \mathcal S(\hat{F}_1;\, 2, 0)} \rho_{0.5}((X^{{F}} -5)_+ \wedge 5), \end{align*}

and

\begin{align*} \sup_{F \in \mathcal S(\hat{F}_2;\, 2, 1.9)} \rho_{0.5}((X^{{F}} -5)_+ \wedge 5) = 2.5647 \gt 1.4748 = \rho_{0.5}((X^{\hat{F}_2}-5)_+ \wedge 5) = \sup_{F \in \mathcal S(\hat{F}_2;\, 2, 0)} \rho_{0.5}((X^{{F}} -5)_+ \wedge 5). \end{align*}

Sharp upper bounds for the premiums of reinsurance contracts, or the worst-case premiums, are helpful for reinsurers in pricing reinsurance contracts when the distribution of the underlying insurance loss is uncertain or unknown.

Besides the applications illustrated above, at the end of this section, we point out that other possible applications of the results derived in Sections 3 and 4, as well as a connection between the following two problems

(4.2) \begin{equation}\sup_{F \in \mathcal{S}} \rho(\ell(X^F)),\end{equation}

and

(4.3) \begin{equation}\min_{\ell \in \mathcal{I}} \sup_{F \in \mathcal{S}} \rho(R_{\ell} (X^F) + \pi(\ell(X^F))),\end{equation}

where $\ell(x)$ is a loss function, $R_{\ell}(X)=X-\ell(X)$ is the retained loss for the insurer under the reinsurance contract $\ell$ , and $\pi(\ell(X))$ is the reinsurance premium.

The problem (4.3) is a popular topic in robust optimal reinsurance studies; see, for example, Landriault et al. (Reference Landriault, Liu and Shi2025) and references therein. If the regularity conditions in the min-max theorem (see, e.g., Fan, Reference Fan1953) are satisfied by the problem (4.3), then it is equivalent to the following problem:

(4.4) \begin{equation} \sup_{F \in \mathcal{S}} \min_{I \in \mathcal{I}} \rho(R_I (X^F) + \pi(I(X^F))).\end{equation}

For any fixed $F \in \mathcal{S}$ , denote by $I^*(x) = I^*(x, F)$ the optimal solution to the inner problem in (4.4), if such a solution exists. Thus, the problem (4.4) further reduces to:

(4.5) \begin{equation} \sup_{F \in \mathcal{S}} \rho(R_{I^*} (X^F) + \pi(I^*(X^F))),\end{equation}

which can be viewed as generalization of the problem (4.2) with $\ell(x) = R_{I^*}(x) + \pi(I^*(x))$ .

Using the above idea and framework (4.5), Boonen and Jiang (Reference Boonen and Jiang2025) solve the problem (4.3) for the case where $\mathcal{S} = \mathcal{S}(\hat{F};\, k, \varepsilon)$ , $\rho$ is a convex distortion risk measure, $I$ satisfies the incentive compatibility condition, $\mathcal I$ is a compact set, and $X$ is a bounded random variable.

However, the equivalence between problems (4.3) and (4.4) holds only if the order of $\min$ and $\sup$ can be exchanged. Solutions to the problem (4.5) heavily rely on the solutions to the inner problem in (4.4). In general, the objective functions in problems (4.2) and (4.5) differ, requiring different methodologies to solve them. This distinction is illustrated in our paper and Cai et al. (Reference Cai, Liu and Yin2024) for the problem (4.2), compared to the approach used by Boonen and Jiang (Reference Boonen and Jiang2025) for the problem (4.5). We believe that investigating problems (4.2), (4.3), and (4.5) with various uncertainty sets $\mathcal S$ and different functions $\ell$ is a valuable topic for future research.