Relative Arbitrage Opportunities in $N$ Investors and Mean-Field Regimes.

This paper analyzes the market behavior and optimal investment strategies to attain relative arbitrage both in the $N$ investors and mean field regimes. An investor competes with a benchmark of market and peer investors, expecting to outperform the benchmark and minimizing the %proportion of initial capital. With market price of risk processes depending on the stock market and investors respectively, the minimal initial capital required is the optimal cost in the $N$-player games and mean field games. It can be characterized as the smallest nonnegative continuous solution of a Cauchy problem. The measure flow of wealth appears in the cost, while the joint flow of wealth and strategy is in state processes. We modify the extended mean field game with common noise and its notion of the uniqueness of Nash equilibrium. There is a unique equilibrium in $N$-player games and mean field games with mild conditions on the equity market.