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

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  • Reinhard Höpfner

    (Universität Mainz)

Abstract

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 .

Suggested Citation

  • Reinhard Höpfner, 2021. "Polynomials under Ornstein–Uhlenbeck noise and an application to inference in stochastic Hodgkin–Huxley systems," Statistical Inference for Stochastic Processes, Springer, vol. 24(1), pages 35-59, April.
  • Handle: RePEc:spr:sistpr:v:24:y:2021:i:1:d:10.1007_s11203-020-09226-0
    DOI: 10.1007/s11203-020-09226-0
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    References listed on IDEAS

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    1. Brice Franke & Thomas Kott, 2013. "Parameter estimation for the drift of a time inhomogeneous jump diffusion process," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 67(2), pages 145-168, May.
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