论文标题
关于扩张单调风险措施的扩展特性
On the extension property of dilatation monotone risk measures
论文作者
论文摘要
令$ \ mathcal {x} $为$ l^1 $的子集，其中包含简单的随机变量的空间$ \ mathcal {l} $和$ρ：\ Mathcal {x} \ rightArow（ - \ iffty，\ infty，\ infty，\ infty] $与FATOU属性相关，我们在$中均在$上张贴。 $σ（l^1，\ Mathcal {l}）$降低半连续和扩张单调功能$ \ OVERLINEρ：l^1 \ rightarrow（ - \ infty，\ infty，\ infty] $。此外，$ \overlineρ$preserlineρ$ preserves preserves honotocitione（quasi）convexity convexity convexity invexity invexity coste corning corning in cashing $ $ $ $ $。 Quasiconvex法律不变功能在[17,20]中证明是我们结果的应用，我们表明，在Orlicz Hearts上的规范风险措施允许自然扩展到$ l^1 $，保留了[4,6]中获得的强大表示形式。
Let $\mathcal{X}$ be a subset of $L^1$ that contains the space of simple random variables $\mathcal{L}$ and $ρ: \mathcal{X} \rightarrow (-\infty,\infty]$ a dilatation monotone functional with the Fatou property. In this note, we show that $ρ$ extends uniquely to a $σ(L^1,\mathcal{L})$ lower semicontinuous and dilatation monotone functional $\overlineρ: L^1 \rightarrow (-\infty,\infty]$. Moreover, $\overlineρ$ preserves monotonicity, (quasi)convexity, and cash-additivity of $ρ$. Our findings complement recent extension results for quasiconvex law-invariant functionals proved in [17,20]. As an application of our results, we show that transformed norm risk measures on Orlicz hearts admit a natural extension to $L^1$ that retains the robust representations obtained in [4,6].
