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Mean-Field Game Formulation for Competitive Market Making: Hamilton-Jacobi-Bellman and Fokker-Planck Equations

Naoki Takata
January 14, 2026

Abstract

We present a rigorous mathematical formalization of the mean-field game framework for competitive market making. The model captures the strategic interaction among a continuum of market makers who compete through their quoted bid and ask prices. We derive the Hamilton-Jacobi-Bellman (HJB) equation governing the value function of a representative agent, the Fokker-Planck equation describing the evolution of the inventory distribution, and the master equation characterizing the full mean-field equilibrium. Optimal quote strategies and arrival intensities are derived in closed form.

Proof

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Timestamp: January 14, 2026 (date recorded on the blockchain)

Formal Verification

All theorems are formalized in Lean 4 (Mathlib commit f897ebcf72cd16f89ab4577d0c826cd14afaafc7). The verification used Aristotle, an automated theorem proving system for Lean 4.

Contact

Naoki Takata
X: @naokitakata
Email: [ntakata [at] proton.me]

This is a preprint. Comments are welcome!