Abstract
This paper has a further study on the insurer’s most unfavorable deductibles and coverage limits for interdependent random losses in the context of the zero-utility premium. For multiple random losses equipped with Archimedean copulas, we conduct stochastic comparison on the insurer’s most unfavorable deductibles and most unfavorable coverage limits, respectively. Also, we build the most unfavorable deductibles for risk-averse insurers. The main results extend and complement those related ones in the literature.
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Notes
A function \(\varphi \) defined on \((0,+\infty )\) is said to be n-monotone if it is differentiable there up to the order \(n-2\) and the derivatives satisfy \((-1)^k\varphi ^{(k)}(t)\ge 0\), for \(k=1,\ldots ,n-2\) and any \(t\in (0,+\infty )\), and further if \((-1)^{(n-2)}\varphi ^{(n-2)}(t)\) is decreasing and convex; here \(\varphi ^{(0)}(t)=\varphi (t)\) and \(\varphi ^{(k)}(t)\) be the kth order derivative evaluated at t.
A random vector \(\varvec{X}\) is said to be positively lower orthant dependent (PLOD) if
$$\begin{aligned} {\mathrm {P}}\bigg (\bigcap _{i=1}^n\{X_i\le x_i\}\bigg )\ge \prod _{i=1}^n{\mathrm {P}}(X_i\le x_i), \qquad \text{ for } \text{ all } x_i, i=1,\ldots ,n. \end{aligned}$$For more we refer readers to Denuit et al. (2005).
A random vector \(\varvec{X}\in {\mathbb {R}}^n\) is comonotonic if \({\mathrm {P}}(\varvec{X}\in B)=1\) for some comonotone subset \(B\in {\mathbb {R}}^n\) such that either \(x_i\le y_i\) or \(x_i\ge y_i\) for \(i=1,\ldots ,n\) whenever both \((x_1,\ldots ,x_n)\) and \((y_1,\ldots ,y_n)\) in B. \(\varvec{X}\) with \(X_1\le _\text {st}\cdots \le _\text {st}X_n\).
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Dr. Yinping You’s research is supported by the National Natural Science Foundation of China (11526093).
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You, Y., Li, X. Most unfavorable deductibles and coverage limits for multiple random risks with Archimedean copulas. Ann Oper Res 259, 485–501 (2017). https://doi.org/10.1007/s10479-017-2537-9
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DOI: https://doi.org/10.1007/s10479-017-2537-9