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Adaptive nonparametric drift estimation for diffusion processes using Faber–Schauder expansions

Author

Listed:
  • Frank Meulen

    (TU Delft)

  • Moritz Schauer

    (Leiden University)

  • Jan Waaij

    (Korteweg-de Vries Institute for Mathematics)

Abstract

We consider the problem of nonparametric estimation of the drift of a continuously observed one-dimensional diffusion with periodic drift. Motivated by computational considerations, van der Meulen et al. (Comput Stat Data Anal 71:615–632, 2014) defined a prior on the drift as a randomly truncated and randomly scaled Faber–Schauder series expansion with Gaussian coefficients. We study the behaviour of the posterior obtained from this prior from a frequentist asymptotic point of view. If the true data generating drift is smooth, it is proved that the posterior is adaptive with posterior contraction rates for the $$L_2$$ L 2 -norm that are optimal up to a log factor. Contraction rates in $$L_p$$ L p -norms with $$p\in (2,\infty ]$$ p ∈ ( 2 , ∞ ] are derived as well.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:sistpr:v:21:y:2018:i:3:d:10.1007_s11203-017-9163-7
    DOI: 10.1007/s11203-017-9163-7
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    References listed on IDEAS

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    1. Weining Shen & Subhashis Ghosal, 2015. "Adaptive Bayesian Procedures Using Random Series Priors," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(4), pages 1194-1213, December.
    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. 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.
    4. 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.
    5. Strauch, Claudia, 2015. "Sharp adaptive drift estimation for ergodic diffusions: The multivariate case," Stochastic Processes and their Applications, Elsevier, vol. 125(7), pages 2562-2602.
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