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Posterior consistency via precision operators for Bayesian nonparametric drift estimation in SDEs

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  • Pokern, Y.
  • Stuart, A.M.
  • van Zanten, J.H.

Abstract

We study a Bayesian approach to nonparametric estimation of the periodic drift function of a one-dimensional diffusion from continuous-time data. Rewriting the likelihood in terms of local time of the process, and specifying a Gaussian prior with precision operator of differential form, we show that the posterior is also Gaussian with the precision operator also of differential form. The resulting expressions are explicit and lead to algorithms which are readily implementable. Using new functional limit theorems for the local time of diffusions on the circle, we bound the rate at which the posterior contracts around the true drift function.

Suggested Citation

  • Pokern, Y. & Stuart, A.M. & van Zanten, J.H., 2013. "Posterior consistency via precision operators for Bayesian nonparametric drift estimation in SDEs," Stochastic Processes and their Applications, Elsevier, vol. 123(2), pages 603-628.
    • Handle: RePEc:eee:spapps:v:123:y:2013:i:2:p:603-628
    DOI: 10.1016/j.spa.2012.08.010
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    References listed on IDEAS

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    1. Federico M. Bandi & Peter C. B. Phillips, 2003. "Fully Nonparametric Estimation of Scalar Diffusion Models," Econometrica, Econometric Society, vol. 71(1), pages 241-283, January.
    2. Harry Van Zanten, 2003. "On Uniform Laws of Large Numbers for Ergodic Diffusions and Consistency of Estimators," Statistical Inference for Stochastic Processes, Springer, vol. 6(2), pages 199-213, May.
    3. Omiros Papaspiliopoulos & Yvo Pokern & Gareth O. Roberts & Andrew M. Stuart, 2012. "Nonparametric estimation of diffusions: a differential equations approach," Biometrika, Biometrika Trust, vol. 99(3), pages 511-531.
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    Cited by:

    1. Frank Meulen & Moritz Schauer & Jan Waaij, 2018. "Adaptive nonparametric drift estimation for diffusion processes using Faber–Schauder expansions," Statistical Inference for Stochastic Processes, Springer, vol. 21(3), pages 603-628, October.
    2. van der Meulen, Frank & Schauer, Moritz & van Zanten, Harry, 2014. "Reversible jump MCMC for nonparametric drift estimation for diffusion processes," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 615-632.
    3. Fabian Dunker & Thorsten Hohage, 2014. "On parameter identification in stochastic differential equations by penalized maximum likelihood," Papers 1404.0651, arXiv.org.

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