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Reversible jump MCMC for nonparametric drift estimation for diffusion processes

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  • van der Meulen, Frank
  • Schauer, Moritz
  • van Zanten, Harry

Abstract

In the context of nonparametric Bayesian estimation a Markov chain Monte Carlo algorithm is devised and implemented to sample from the posterior distribution of the drift function of a continuously or discretely observed one-dimensional diffusion. The drift is modeled by a scaled linear combination of basis functions with a Gaussian prior on the coefficients. The scaling parameter is equipped with a partially conjugate prior. The number of basis functions in the drift is equipped with a prior distribution as well. For continuous data, a reversible jump Markov chain algorithm enables the exploration of the posterior over models of varying dimension. Subsequently, it is explained how data-augmentation can be used to extend the algorithm to deal with diffusions observed discretely in time. Some examples illustrate that the method can give satisfactory results. In these examples a comparison is made with another existing method as well.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:csdana:v:71:y:2014:i:c:p:615-632
    DOI: 10.1016/j.csda.2013.03.002
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    References listed on IDEAS

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    1. Alexandros Beskos & Omiros Papaspiliopoulos & Gareth O. Roberts & Paul Fearnhead, 2006. "Exact and computationally efficient likelihood‐based estimation for discretely observed diffusion processes (with discussion)," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(3), pages 333-382, June.
    2. 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.
    3. repec:dau:papers:123456789/1908 is not listed on IDEAS
    4. Eraker, Bjorn, 2001. "MCMC Analysis of Diffusion Models with Application to Finance," Journal of Business & Economic Statistics, American Statistical Association, vol. 19(2), pages 177-191, April.
    5. Håvard Rue & Sara Martino & Nicolas Chopin, 2009. "Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 319-392, April.
    6. 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 Waaij, Jan & van Zanten, Harry, 2017. "Full adaptation to smoothness using randomly truncated series priors with Gaussian coefficients and inverse gamma scaling," Statistics & Probability Letters, Elsevier, vol. 123(C), pages 93-99.

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