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We present a model for price dynamics in the Automated Market Makers (AMM)setting. Within this framework, we propose a reference market price following ageometric Brownian motion. The AMM price is constrained by upper and lowerbounds, determined by constant multiplications of the reference price. Throughthe utilization of local times and excursion-theoretic approaches, we deriveseveral analytical results, including its time-changed representation andlimiting behavior.

An arbitrage driven price dynamics of Automated Market Makers in the presence of fees