A mathematical framework for modelling CLMM dynamics in continuous time

This paper develops a continuous-time economic model to analyze the dynamics of Concentrated Liquidity Market Makers (CLMMs), specifically Uniswap V3 and the strategic behavior of arbitrageurs. We model the evolution of liquidity profiles as measure-valued processes, enabling a precise characterization of how liquidity distributions respond to trading activity. Our analysis reveals that providing concentrated liquidity is economically equivalent to a covered call strategy, where liquidity providers effectively sell the time premium of an implicit call option in exchange for trading fees. Consequently, we derive an endogenous expression for equilibrium liquidity provision, demonstrating that it decreases with the time premium and increases with the expected time the asset price spends within a given interval. To address price discovery, we solve the optimization problems for three distinct arbitrage regimes: finite-horizon, discounted infinite-horizon, and ergodic control. For each scenario, we derive closed-form solutions for optimal trading strategies, illustrating how arbitrageurs maintain price alignment between decentralized and centralized venues.