  Automated Market Makers (AMMs) are essential in Decentralized Finance (DeFi)
as they match liquidity supply with demand. They function through liquidity
providers (LPs) who deposit assets into liquidity pools. However, the asset
trading prices in these pools often trail behind those in more dynamic,
centralized exchanges, leading to potential arbitrage losses for LPs. This
issue is tackled by adapting market maker bonding curves to trader behavior,
based on the classical market microstructure model of Glosten and Milgrom. Our
approach ensures a zero-profit condition for the market maker's prices. We
derive the differential equation that an optimal adaptive curve should follow
to minimize arbitrage losses while remaining competitive. Solutions to this
optimality equation are obtained for standard Gaussian and Lognormal price
models using Kalman filtering. A key feature of our method is its ability to
estimate the external market price without relying on price or loss oracles. We
also provide an equivalent differential equation for the implied dynamics of
canonical static bonding curves and establish conditions for their optimality.
Our algorithms demonstrate robustness to changing market conditions and
adversarial perturbations, and we offer an on-chain implementation using
Uniswap v4 alongside off-chain AI co-processors.
