DeFi Research · Stochastic Control
A research-driven dashboard that formalises AMM invariant curvature as a stochastic control variable — simulating impermanent loss, fee-compensation equilibrium, and utility-maximising liquidity provision via the HJB framework.
HJB
Control Framework
IL
Impermanent Loss
AMM
Invariant Design
LP
Utility Optimisation
Automated Market Makers like Uniswap use fixed invariant curves (x·y=k) that don't adapt to market conditions. Liquidity providers suffer impermanent loss when prices diverge from the pool's implied rate — a cost that existing AMM designs accept as fixed rather than treating it as a control variable to be minimised.
The project formalises AMM invariant curvature as a stochastic control variable in continuous time. The liquidity provider's problem is formulated as a utility maximisation problem: choose the invariant curvature path that maximises expected utility subject to the price dynamics of the risky asset.
The Hamilton-Jacobi-Bellman equation characterises the value function of this control problem. Numerical solutions are computed and visualised — showing how the optimal curvature responds to price volatility, fee rates, and LP risk aversion. Fee-compensation equilibrium conditions are derived analytically and verified numerically.
The prototype demonstrates that treating curvature as a dynamic control variable — rather than a fixed constant — can meaningfully reduce impermanent loss for risk-averse LPs under certain fee and volatility regimes. The full mathematical derivation and interactive simulations are documented in the companion report.