共同基金定理在哪些金融市场中成立？
In which Financial Markets do Mutual Fund Theorems hold true?
摘要：
研究了有限时间内一般半鞅金融市场S中的共同基金定理，其中代理人最大化终端财富的期望效用.建立了：1）设N是数字投资组合（即对数效用最优投资组合）的财富过程。如果任意一个到期日为T的路径无关期权可以通过在N中交易\emph{only}来复制，则（MFT）对一般效用函数成立，并且该数字资产组合可以作为共同基金，这推广了Merton关于Black-Scholes市场的经典结果.相反，在一个补充的弱完备性假设下，我们证明了（MFT）对一般效用函数的有效性意味着上述数字投资组合上的期权具有相同的可复制性。2)如果对于给定的一类效用函数（即投资者），（MFT）在所有完备布朗金融市场S中成立，则所有投资者使用相同的效用函数U，它必须是HARA型。这是卡斯和斯蒂格利茨经典作品精神的结果。
英文摘要：
The Mutual Fund Theorem (MFT) is considered in a general semimartingale financial market S with a finite time horizon T, where agents maximize expected utility of terminal wealth. It is established that: 1) Let N be the wealth process of the numéraire portfolio (i.e. the optimal portfolio for the log utility). If any path-independent option with maturity T written on the numéraire portfolio can be replicated by trading \emph{only} in N, then the (MFT) holds true for general utility functions, and the numéraire portfolio may serve as mutual fund. This generalizes Merton's classical result on Black-Scholes markets. Conversely, under a supplementary weak completeness assumption, we show that the validity of the (MFT) for general utility functions implies the same replicability property for options on the numéraire portfolio described above. 2) If for a given class of utility functions (i.e. investors) the (MFT) holds true in all complete Brownian financial markets S, then all investors use the same utility function U, which must be of HARA type. This is a result in the spirit of the classical work by Cass and Stiglitz.
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