Risk control is one of the crucial problems in finance. One of the most common ways to mitigate risk of an investor’s financial position is to set up a portfolio of hedging securities whose aim is to absorb unexpected losses and thus provide the investor with an acceptable level of security. In this respect, it is clear that investors will try to reach acceptability at the lowest possible cost. Mathematically, this problem leads naturally to considering set-valued maps that associate to each financial position the corresponding set of optimal hedging portfolios, i.e., of portfolios that ensure acceptability at the cheapest cost. Among other properties of such maps, the ability to ensure lower semicontinuity and continuous selections is key from an operational perspective. It is known that lower semicontinuity generally fails in an infinite-dimensional setting. In this note, we show that neither lower semicontinuity nor, more surprisingly, the existence of continuous selections can be a priori guaranteed even in a finite-dimensional setting. In particular, this failure is possible under arbitrage-free markets and convex risk measures.

中文翻译：

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在凸风险度量下并不总是会持续选择最优投资组合

风险控制是财务中的关键问题之一。减轻投资者财务状况风险的最常见方法之一是建立对冲证券投资组合，其目的是吸收意外损失，从而为投资者提供可接受的担保水平。在这方面，很明显，投资者将试图以尽可能低的成本获得认可。从数学上讲，此问题自然导致考虑将与每个财务状况相关联的最佳套期保值投资组合（即以最低成本确保可接受性的投资组合）的集合值映射。从操作的角度来看，确保此类图的其他特性中，确保较低的半连续性和连续选择的能力至关重要。众所周知，下半连续性通常会在无穷维设置中失效。在此注释中，我们表明，即使在有限维设置中，也不能先验地保证较低的半连续性或更令人惊讶的是，连续选择的存在。特别是，在无套利市场和凸风险措施下，这种失败是可能的。