Dynamics of the Liquidity Profile in Concentrated Liquidity Provision: An SPIDE Approach
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 L2 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.
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BibTeX
@misc{lee2026dynamics,
title = {{Dynamics of the Liquidity Profile in Concentrated Liquidity Provision: An SPIDE Approach}},
author = {Cheuk Yin Lee and Jimmy Risk and Shen-Ning Tung and Tai-Ho Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30539},
eprint = {2610.30539},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30539},
}
RIS
TY - UNPB
TI - Dynamics of the Liquidity Profile in Concentrated Liquidity Provision: An SPIDE Approach
AU - Cheuk Yin Lee
AU - Jimmy Risk
AU - Shen-Ning Tung
AU - Tai-Ho Wang
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.30539
UR - https://arxiv.org/abs/2610.30539
AB - 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.
ER -