| Abstract: |
| We consider relative arbitrage opportunities in a market with competitive investors through stochastic differential games in the limit as the number of players tends to infinity. We explore a conditional McKean-Vlasov system to study the market dynamics coupled to the expected trading volume of investors, and characterize optimal arbitrage in terms of the volatilities. Here, the mean-field interaction is considered through a joint distribution of wealth and strategies. In this setting, the optimal relative arbitrage constitutes the strong equilibrium of an extended mean-field game. We provide conditions for the existence and uniqueness of the mean-field equilibrium. We further prove the propagation of chaos result for the finite-player game counterpart, and demonstrate that the Nash equilibrium converges to the mean field equilibrium when the population grows to infinity. |
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