Drift coefficient ($b$)

The drift coefficient is the coefficient $b$ of $dt$ in a stochastic differential equation. It describes the local deterministic tendency of the process.

For continuous observations of an Itô diffusion $d{X}_t=b(X_t)dt+\sigma (X_t)d{W}_t$, a local drift estimator at $x$ ison $\{A_T>0\}$. Its error is the average of $b(X_t)-b(x)$ over that window, plus $A_T^{-1}{\int }_0^T{1}_{\{|X_t-x|\le h\}}\sigma (X_t)d{W}_t$. A local Hölder continuity bound with exponent $\alpha$ bounds the first term by $L{h}^{\alpha }$.

The commonly used noise scale $(Th{)}^{-1/2}$ requires occupation control. For a stationary model with continuous positive density $\pi (x)$ of its invariant distribution and locally bounded $\sigma$, the Itô isometry makes the numerator's second moment $O(Th)$. If additionally $A_T/(2Th\pi (x))\to 1$ in probability in the chosen long-time, shrinking-window regime, division gives stochastic error $O_{\mathbb{P}}((Th{)}^{-1/2})$. Positivity of the invariant density alone does not establish that concentration of the random denominator.