Derivation: Stochastic Taylor Expansion and the Milstein Scheme

The Euler scheme truncates something — but truncates what? The answer is the stochastic (Itô–)Taylor expansion: just as deterministic integrators come from Taylor's theorem with integral remainder, SDE integrators come from iterating Itô's lemma into itself. Carrying the expansion one term further than Euler produces Milstein. We follow L3's route: first the deterministic warm-up, then the stochastic case, then the explicit computation of the key double integral $I_{(1,1)}$.