Paper: arXiv 2610.30539

Authors: Cheuk Yin Lee, Jimmy Risk, Shen-Ning Tung, Tai-Ho Wang

Abstract

We model the dynamics of the liquidity profile in Uniswap~V3 and other concentrated-liquidity automated market makers with a stochastic partial integro-differential equation (SPIDE). The profile, the aggregate of all provider positions across tick space, evolves under a degenerate Sturm–Liouville operator, which encodes mean-reverting concentration around the market price, coupled with a Hilbert–Schmidt integral operator, which captures nonlocal redistribution driven by provider repositioning. We establish existence and uniqueness of the mild solution in a weighted $L^2$ space built on the Gegenbauer eigensystem, and reduce the SPIDE to a coupled system of linear stochastic differential equations whose low-order modes describe the level, slope, and curvature of the profile. Pairing the SPIDE with a log-price process yields tractable expressions for pool reserves, pool value, and loss-versus-rebalancing under Ornstein–Uhlenbeck-type and Black–Scholes-type noise. A finite-rank calibration in a fixed reference frame, applied to three Uniswap pools, gives interpretable estimates and lowers active-band forecast error relative to a martingale benchmark on the two pools compatible with the stationary specification. On the third pool, out-of-sample diagnostics are consistent with nonstationarity, which motivates price-aware and time-varying extensions.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a highly complex mathematical framework using SPIDEs and Sturm-Liouville operators to model liquidity. It supports this with empirical calibration on real Uniswap V3 data, demonstrating strong rigor in its application and validation. The approach is highly novel, tackling a complex problem in DeFi with advanced mathematical tools.

Research Flowchart

  flowchart TD
    A[Research Goal: Model Liquidity Profile Dynamics in Concentrated AMMs] --> B{Key Methodology: SPIDE Formulation};
    B -- Stochastic Partial Integro-Differential Equation --> C[SPIDE Components: Degenerate Sturm-Liouville Operator (Concentration) + Hilbert-Schmidt Integral Operator (Repositioning)];
    C -- Mathematical Framework --> D{Computational Process: Gegenbauer Eigensystem & SDE Reduction};
    D -- Inputs: Log-Price Process (OU/Black-Scholes Noise) & Uniswap Pool Data --> E[Key Findings: Existence/Uniqueness of Mild Solution & Tractable Expressions for Pool Metrics];
    E -- Calibration & Forecast --> F{Outcomes: Interpretable Estimates, Lower Forecast Error (stationary pools), Identification of Non-Stationarity};