We consider a single-period portfolio selection problem for an investor, maximizing the expected ratio of the portfolio utility to the utility of a best asset taken in hindsight. The decision rules are based on the history of stock returns with unknown distribution. Assuming that the utility function is Lipschitz or Hölder continuous (the concavity is not required), we obtain high probability utility bounds under the sole assumption that the returns are independent and identically distributed. These bounds depend only on the utility function, the number of assets and the number of observations. For concave utilities similar bounds are obtained for the portfolios produced by the exponentiated gradient algorithm. Also we use statistical experiments to study risk and generalization properties of empirically optimal portfolios. Herein we consider a model with one risky asset and several datasets, containing real stock prices.

我们考虑了投资者的单期投资组合选择问题，从而最大化了事后发现的投资组合效用与最佳资产效用的预期比率。决策规则基于具有未知分布的股票收益的历史记录。假设效用函数是Lipschitz或Hölder连续的（不需要凹度），则在收益是独立且分布均匀的唯一假设下，我们获得了高概率效用范围。这些界限仅取决于效用函数，资产数量和观察值数量。对于凹面效用，通过指数梯度算法产生的投资组合获得了相似的界限。此外，我们使用统计实验来研究经验最优投资组合的风险和泛化性质。