A mathematical framework for modelling CLMM dynamics in continuous time

ArXiv ID: 2412.18580 “View on arXiv”

Authors: Unknown

Abstract

This paper develops a rigorous mathematical framework for analyzing Concentrated Liquidity Market Makers (CLMMs) in Decentralized Finance (DeFi) within a continuous-time setting. We model the evolution of liquidity profiles as measure-valued processes and characterize their dynamics under continuous trading. Our analysis encompasses two critical aspects of CLMMs: the mechanics of concentrated liquidity provision and the strategic behavior of arbitrageurs. We examine three distinct arbitrage models – myopic, finite-horizon, and infinite-horizon with discounted and ergodic controls – and derive closed-form solutions for optimal arbitrage strategies under each scenario. Importantly, we demonstrate that the presence of trading fees fundamentally constrains the admissible price processes, as the inclusion of fees precludes the existence of diffusion terms in the price process to avoid infinite fee generation. This finding has significant implications for CLMM design and market efficiency.

Keywords: Concentrated Liquidity Market Makers, Decentralized Finance (DeFi), Arbitrage strategies, Liquidity profiles, Market microstructure

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats
  • Why: The paper is extremely dense with advanced mathematics, including measure-valued processes, stochastic analysis, and closed-form solutions for various control problems, placing it at the high end of math complexity. However, it lacks any backtesting, code, or empirical data, focusing purely on theoretical derivations with no practical implementation metrics, resulting in very low empirical rigor.
  flowchart TD
    A["Research Goal<br>Model CLMM dynamics<br>in continuous time"] --> B["Data/Inputs<br>CLMM mechanics &<br>Arbitrage scenarios"]

    B --> C["Methodology<br>Measure-valued processes<br>Continuous-time modeling"]

    C --> D["Computational Processes<br>Three arbitrage models:<br>Myopic, Finite, Infinite horizon"]

    D --> E["Key Findings/Outcomes<br>1. Closed-form optimal strategies<br>2. Fees constrain price processes<br>3. No diffusion terms with fees"]