论文标题
可分解的总和及其对自然质子vex风险措施的影响
Decomposable sums and their implications on naturally quasiconvex risk measures
论文作者
论文摘要
凸度和准综合性是两种属性，可捕获风险度量的多元化概念。在两者之间，存在天然的准杂种性，一种古老但并不是众所周知的属性，比凸性弱，但比ic iciconvexity强。关于自然准杂种性的详细讨论仍然缺少，本文旨在在有条件的风险措施的情况下填补这一空白。我们将天然的准蔬菜与可添加分解的总和联系起来。在1980年代定义的有限维矢量空间定义的凸索指数的概念在可分解总和的讨论中起着至关重要的作用。我们提出了拓扑矢量空间中凸指数的一般处理，并使用它来研究自然的准风险措施。我们证明，在轻度的连续性和当地条件下，天然的准清理和凸度对于$ l^p $空间，$ p \ geq 1 $的有条件风险度量等效。最后，我们讨论了相对于$ l^2 $的正常基础的替代概念。
Convexity and quasiconvexity are two properties that capture the concept of diversification for risk measures. Between the two, there is natural quasiconvexity, an old but not so well-known property weaker than convexity but stronger than quasiconvexity. A detailed discussion on natural quasiconvexity is still missing and this paper aims to fill this gap in the setting of conditional risk measures. We relate natural quasiconvexity to additively decomposable sums. The notion of convexity index, defined in 1980s for finite-dimensional vector spaces, plays a crucial role in the discussion of decomposable sums. We propose a general treatment of convexity index in topological vector spaces and use it to study naturally quasiconvex risk measures. We prove that natural quasiconvexity and convexity are equivalent for conditional risk measures on $L^p$ spaces, $p \geq 1$, under mild continuity and locality conditions. Finally, we discuss an alternative notion of locality with respect to an orthonormal basis in $L^2$.
