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We discuss estimation problems where a polynomial    $$s\rightarrow \sum _{i=0}^\ell \vartheta _i s^i$$  s → ∑ i = 0 ℓ ϑ i s i with strictly positive leading coefficient is observed under Ornstein–Uhlenbeck noise over a long time interval. We prove local asymptotic normality (LAN) and specify asymptotically efficient estimators. We apply this to the following problem: feeding noise    $$dY_t$$  d Y t into the classical (deterministic) Hodgkin–Huxley model in neuroscience, with    $$Y_t=\vartheta t + X_t$$  Y t = ϑ t + X t and  X  some Ornstein–Uhlenbeck process with backdriving force    $$\tau $$  τ , we have asymptotically efficient estimators for the pair    $$(\vartheta ,\tau )$$  ( ϑ , τ ) ; based on observation of the membrane potential up to time  n , the estimate for    $$\vartheta $$  ϑ converges at rate    $$\sqrt{n^3\,}$$  n 3 .

Polynomials under Ornstein–Uhlenbeck noise and an application to inference in stochastic Hodgkin–Huxley systems