自然拟凸风险测度的可分解和及其意义
Decomposable sums and their implications on naturally quasiconvex risk measures
摘要：
凸性和拟凸性是描述风险度量分散化概念的两个性质。在这两者之间，有一个自然的拟凸性，这是一个古老但不太知名的性质，比凸性弱，但比拟凸性强。自然拟凸性的详细讨论仍然缺失，本文旨在填补这一空白的条件风险措施的设置。我们将自然拟凸性与可加可分解和联系起来。20世纪80年代在有限维向量空间中定义的凸性指数的概念在可分解和的讨论中起着至关重要的作用。提出了拓扑向量空间中凸性指标的一般处理方法，并利用它研究了自然拟凸风险测度。在弱连续性和局部性条件下，证明了$L^p$空间$p \geq 1$上条件风险测度的自然拟凸性和自然凸性是等价的.最后，我们讨论了另一种概念的局部性相对于一个正交基在$L^2$。
英文摘要：
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 $L^2$.
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