Drift estimator (${\hat{b}}_T(x,h)$)

From a continuously observed Itô diffusion, a local drift estimator divides ${\int }_0^T{1}_{[x-h,x+h]}(X_t)d{X}_t$ by the corresponding diffusion occupation time. Its error separates into a local Hölder continuity bias and an Itô integral divided by occupation time. The Itô isometry gives a pointwise noise scale $(Th{)}^{-1/2}$ when the invariant density is locally positive.