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The Fatou property of law-invariant risk measures

Made Tantrawan and Denny H. Leung

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Abstract: This paper presents several results on the Fatou property of quasiconvex law-invariant functionals defined on a rearrangement invariant space $\mathcal{X}$. First, we show that for any proper quasiconvex law-invariant functional $\rho$ on $\mathcal{X}$, the Fatou property, $\sigma(\mathcal{X},\mathcal{X}_n^\sim)$-lower semicontinuity and $\sigma(\mathcal{X},L^\infty)$-lower semicontinuity of $\rho$ are equivalent, where $\mathcal{X}_n^\sim$ is the order continuous dual of $\mathcal{X}$. Second, we provide some relations between the Fatou property and some other types of lower semicontinuity, namely norm lower semicontinuity and the strong Fatou property. In particular, we show that when $\mathcal{X}\neq L^1$, the strong Fatou property and the Fatou property coincide. Finally, we generalize some extension results by Gao et al. [9] to a general rearrangement invariant space.

Date: 2018-10
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Handle: RePEc:arx:papers:1810.10374