Adaptive Milstein Methods for Diffusions with Discontinuous Drift

We apply an adaptive interface method to a two-dimensional diffusion with finite-speed rebalancing outside an elliptical no-trade region. An explicit change of variables removes the drift jump, preserves commutativity of the noise, and admits a quantitative global inverse certificate. The resulting approximation has terminal root-mean-square error $O(\delta)$ at expected Brownian observation cost $O(\delta^{-1})$. Terminal quadratic tracking risk has $L^1$ error $O(\delta)$; a randomized within-step estimator of gross turnover has $L^1$ error $O(\delta^{2/3})$. A synthetic experiment illustrates the risk–turnover trade-off but finds no runtime advantage over ordinary Euler at the tested tolerances. The complete underlying interface proof is included in the proof sections below. Independent proof review and a wider originality assessment remain outstanding.