# Relative Arbitrage Opportunities in an Extended Mean Field System

arXiv: https://arxiv.org/abs/2311.02690 · HTML: https://mlsanity.com/paper/2311.02690/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Nov 2023.

## Built on by (most cited first, 3 of 3)

- [End-to-End Training of High-Dimensional Optimal Control with Implicit Hamiltonians via Jacobian-Free Backpropagation](https://mlsanity.com/paper/2510.00359/) (2025)
- [Pathwise Learning of Stochastic Dynamical Systems with Partial Observations](https://mlsanity.com/paper/2601.21860/) (2026)
- [On the Convergence of Jacobian-Free Backpropagation for Optimal Control Problems with Implicit Hamiltonians](https://mlsanity.com/paper/2602.00921/) (2026)

## Abstract (arXiv)

This paper studies relative arbitrage opportunities in a market with competitive investors through stochastic differential games in the limit as the number of players tends to infinity. With common noises introduced by the stock capitalization processes, we establish a conditional McKean-Vlasov system to study the market dynamics coupled to the expected trading volume of investors. We show that optimal arbitrage can be characterized as a solution of a Cauchy PDE constructed by the volatility terms in the market model. The structure of the market dynamics can be relaxed, and we provide a theoretical framework to study a general mean-field system, where the interaction is characterized by 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.