We analyze a family of portfolio management problems under relative
performance criteria, for fund managers having CARA or CRRA utilities and
trading in a common investment horizon in log-normal markets. We construct
explicit constant equilibrium strategies for both the finite population games
and the corresponding mean field games, which we show are unique in the class
of constant equilibria. In the CARA case, competition drives agents to invest
more in the risky asset than they would otherwise, while in the CRRA case
competitive agents may over- or under-invest, depending on their levels of risk
tolerance.

研究了相对绩效标准下的资产组合管理问题，考虑了基金经理在对数正态市场中交易的情况，推导出了 CARA 和 CRRA 情形下的平衡策略，发现竞争促使 Agent 更多地投资于风险资产，但具体投资量取决于其风险承受能力。

基于相对表现指标的均场和 n 智能体投资优化博弈

Mean field and n-agent games for optimal investment under relative  performance criteria

Portfolio management is the decision-making process of allocating an amount
of fund into different financial investment products. Cryptocurrencies are
electronic and decentralized alternatives to government-issued money, with
Bitcoin as the best-known example of a cryptocurrency. This paper presents a
model-less convolutional neural network with historic prices of a set of
financial assets as its input, outputting portfolio weights of the set. The
network is trained with 0.7 years' price data from a cryptocurrency exchange.
The training is done in a reinforcement manner, maximizing the accumulative
return, which is regarded as the reward function of the network. Backtest
trading experiments with trading period of 30 minutes is conducted in the same
market, achieving 10-fold returns in 1.8 months' periods. Some recently
published portfolio selection strategies are also used to perform the same
back-tests, whose results are compared with the neural network. The network is
not limited to cryptocurrency, but can be applied to any other financial
markets.

本文提出了一种使用历史价格数据的基于模型的卷积神经网络，输出给定金融资产的投资组合权重。通过强化学习的方式，对加密货币交易所 0.7 年的价格数据进行了训练，最终在 30 分钟内获得了 10 倍的回报，并将其与其他投资组合策略进行了比较。该模型不仅适用于加密货币市场，还可应用于任何其他金融市场。