Strong well-posedness of additive-noise equations with bounded continuous drift

The zero extension of a scale diffusion coefficient at finite endpoints makes the transformed equation globally Lipschitz. Inverting its solution gives the original additive-noise equation. The pathwise bound $|X_t|\le |x|+{\sup }_{s\le T}|W_s|+\Vert b{\Vert }_{\infty }T$ prevents the inverse scale from reaching infinity at any finite time. This proves global strong existence and pathwise uniqueness even when the original drift is not Lipschitz.