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

We discuss estimation problems where a polynomial $$s\rightarrow \sum _{i=0}^\ell \vartheta _i s^i$$ 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 $$dY_t$$ into the classical (deterministic) Hodgkin–Huxley model in neuroscience, with $$Y_t=\vartheta t + X_t$$ and X some Ornstein–Uhlenbeck process with backdriving force $$\tau $$ , we have asymptotically efficient estimators for the pair $$(\vartheta ,\tau )$$ ; based on observation of the membrane potential up to time n, the estimate for $$\vartheta $$ converges at rate $$\sqrt{n^3\,}$$ .