Mean‐ portfolio selection and ‐arbitrage for coherent risk measures
研究单期金融市场中基于一致风险度量的均值-投资组合选择，发现最优组合集非空且紧致，但可能不存在有效组合（即ρ-套利），并给出其充要条件，与等价鞅测度密切相关。
Abstract We revisit mean‐risk portfolio selection in a one‐period financial market where risk is quantified by a positively homogeneous risk measure . We first show that under mild assumptions, the set of optimal portfolios for a fixed return is nonempty and compact. However, unlike in classical mean‐variance portfolio selection, it can happen that no efficient portfolios exist. We call this situation ‐arbitrage, and prove that it cannot be excluded—unless is as conservative as the worst‐case risk measure. After providing a primal characterization of ‐arbitrage, we focus our attention on coherent risk measures that admit a dual representation and give a necessary and sufficient dual characterization of ‐arbitrage. We show that the absence of ‐arbitrage is intimately linked to the interplay between the set of equivalent martingale measures (EMMs) for the discounted risky assets and the set of absolutely continuous measures in the dual representation of . A special case of our result shows that the market does not admit ‐arbitrage for Expected Shortfall at level if and only if there exists an EMM such that .