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We study a family of portfolio management problems under relative performance competition and memory effect. The individual investor’s goal is to maximize her utility via allocating her wealth wisely into an individual stock and a risk-free bond when the historical performance of her own and other investors’ portfolio affects her wealth allocation decision through the dynamics of wealth and the competitive behavior embedded in utility functions. This type of portfolio optimization problems is investigated in both finite-population games and mean-field games in the present paper. Due to the memory effect and relative concern behavior, this type of problems is generally very challenging. We establish certain conditions under which the problem can be simplified to obtain the optimal investment strategies for power and logarithmic (CRRA) type investors playing finite-population and mean-field games. The resulting optimal investment strategies for N-player and mean-field games are analyzed and compared to each other as well as to the classical memoryless and noncompetitive models.

Optimal investment mean-field and N-player games with memory effect and relative performance competition