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The relative arbitrage portfolio, formulated in Stochastic Portfolio Theory(SPT), outperforms a benchmark portfolio over a given time-horizon withprobability one. This paper analyzes the market behavior and optimal investmentstrategies to attain relative arbitrage both in the $N$ investors and meanfield regimes under some market conditions. An investor competes with abenchmark of market and peer investors, expecting to outperform the benchmarkand minimizing the initial capital.  With market price of risk processes depending on the market portfolio andinvestors, we develop a systematic way to solve multi-agent optimizationproblem within SPT's framework. The objective can be characterized by thesmallest nonnegative continuous solution of a Cauchy problem. By a modificationin the structure of the extended mean field game with common noise and itsnotion of the uniqueness of Nash equilibrium, we show a unique equilibrium in$N$-player games and mean field games with mild conditions on the equitymarket.

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