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