Electronic Journal of Statistics

Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals

Frank van der Meulen and Moritz Schauer

Full-text: Open access

Abstract

Estimation of parameters of a diffusion based on discrete time observations poses a difficult problem due to the lack of a closed form expression for the likelihood. From a Bayesian computational perspective it can be casted as a missing data problem where the diffusion bridges in between discrete-time observations are missing. The computational problem can then be dealt with using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown parameters appear in the diffusion coefficient, direct implementation of data-augmentation results in a Markov chain that is reducible. Furthermore, data-augmentation requires efficient sampling of diffusion bridges, which can be difficult, especially in the multidimensional case.

We present a general framework to deal with with these problems that does not rely on discretisation. The construction generalises previous approaches and sheds light on the assumptions necessary to make these approaches work. We define a random-walk type Metropolis-Hastings sampler for updating diffusion bridges. Our methods are illustrated using guided proposals for sampling diffusion bridges. These are Markov processes obtained by adding a guiding term to the drift of the diffusion. We give general guidelines on the construction of these proposals and introduce a time change and scaling of the guided proposal that reduces discretisation error. Numerical examples demonstrate the performance of our methods.

Article information

Source
Electron. J. Statist. Volume 11, Number 1 (2017), 2358-2396.

Dates
Received: July 2016
First available in Project Euclid: 27 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1495850628

Digital Object Identifier
doi:10.1214/17-EJS1290

Zentralblatt MATH identifier
1378.62050

Subjects
Primary: 62M05: Markov processes: estimation 60J60: Diffusion processes [See also 58J65]
Secondary: 62F15: Bayesian inference 65C05: Monte Carlo methods

Keywords
Multidimensional diffusion bridge data augmentation discretisation of path integral linear process innovation process non-centred parametrisation FitzHugh-Nagumo model

Rights
Creative Commons Attribution 4.0 International License.

Citation

van der Meulen, Frank; Schauer, Moritz. Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals. Electron. J. Statist. 11 (2017), no. 1, 2358--2396. doi:10.1214/17-EJS1290. https://projecteuclid.org/euclid.ejs/1495850628


Export citation

References

  • Beskos, A., Papaspiliopoulos, O., Roberts, G. O. and Fearnhead, P. (2006). Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes., J. R. Stat. Soc. Ser. B Stat. Methodol. 68 333–382. With discussions and a reply by the authors.
  • Beskos, A., Roberts, G., Stuart, A. and Voss, J. (2008). MCMC Methods for diffusion bridges., Stochastics and Dynamics 08 319–350.
  • Bezanson, J., Karpinski, S., Shah, V. B. and Edelman, A. (2012). Julia: A Fast Dynamic Language for Technical Computing., CoRR abs/1209.5145.
  • Bladt, M. and Sørensen, M. (2014). Simple simulation of diffusion bridges with application to likelihood inference for diffusions., Bernoulli 20 645–675.
  • Bladt, M. and Sørensen, M. (2015). Simulation of multivariate diffusion bridges., To appear in Journal of the Royal Statistical Society, series B.
  • Chib, S., Pitt, M. K. and Shephard, N. (2004). Likelihood based inference for diffusion driven models Economics Papers No. 2004-W20, Economics Group, Nuffield College, University of, Oxford.
  • Clark, J. M. C. (1990). The simulation of pinned diffusions. In, Decision and Control, 1990., Proceedings of the 29th IEEE Conference on 1418–1420. IEEE.
  • Cotter, S. L., Roberts, G. O., Stuart, A. M. and White, D. (2013). MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster., Statist. Sci. 28 424–446.
  • Delyon, B. and Hu, Y. (2006). Simulation of conditioned diffusion and application to parameter estimation., Stochastic Processes and their Applications 116 1660–1675.
  • Durham, G. B. and Gallant, A. R. (2002). Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes., J. Bus. Econom. Statist. 20 297–338. With comments and a reply by the authors.
  • Elerian, O., Chib, S. and Shephard, N. (2001). Likelihood inference for discretely observed nonlinear diffusions., Econometrica 69 959–993.
  • Eraker, B. (2001). MCMC analysis of diffusion models with application to finance., J. Bus. Econom. Statist. 19 177–191.
  • Fuchs, C. (2013)., Inference for diffusion processes. Springer, Heidelberg With applications in life sciences, With a foreword by Ludwig Fahrmeir.
  • Golightly, A. and Wilkinson, D. J. (2008). Bayesian inference for nonlinear multivariate diffusion models observed with error., Comput. Statist. Data Anal. 52 1674–1693.
  • Golightly, A. and Wilkinson, D. J. (2010)., Learning and Inference in Computational Systems Biology Markov chain Monte Carlo algorithms for SDE parameter estimation, 253–276. MIT Press.
  • Gugushvili, S. and Spreij, P. (2012). Parametric inference for stochastic differential equations: a smooth and match approach., ALEA Lat. Am. J. Probab. Math. Stat. 9 609–635.
  • Gyöngy, I. (1998). A note on Euler’s approximations., Potential Anal. 8 205–216.
  • Jensen, C. Anders (2014)., Statistical Inference for Partially Observed Diffusion Processes. Ph.d. Thesis University of Copenhagen.
  • Jensen, A. C., Ditlevsen, S., Kessler, M. and Papaspiliopoulos, O. (2012). Markov chain Monte Carlo approach to parameter estimation in the FitzHugh-Nagumo model., Phys. Rev. E 86 041114.
  • Khasminskii, R. Z. and Klebaner, F. C. (2001). Long term behavior of solutions of the Lotka-Volterra system under small random perturbations., Ann. Appl. Probab. 11 952–963.
  • Küchler, U. and Sørensen, M. (1997)., Exponential families of stochastic processes. Springer Series in Statistics. Springer-Verlag, New York.
  • Lin, M., Chen, R. and Mykland, P. (2010). On generating Monte Carlo samples of continuous diffusion bridges., J. Amer. Statist. Assoc. 105 820–838.
  • Neal, R. M. (1999). Regression and classification using Gaussian process priors. In, Bayesian statistics, 6 (Alcoceber, 1998) 475–501. Oxford Univ. Press, New York.
  • Papaspiliopoulos, O., Roberts, G. O. and Sköld, M. (2003). Non-centered parameterizations for hierarchical models and data augmentation. In, Bayesian statistics, 7 (Tenerife, 2002) 307–326. Oxford Univ. Press, New York With a discussion by Alan E. Gelfand, Ole F. Christensen and Darren J. Wilkinson, and a reply by the authors.
  • Papaspiliopoulos, O. and Roberts, G. (2012). Importance sampling techniques for estimation of diffusion models. In, Statistical Methods for Stochastic Differential Equations. Monographs on Statistics and Applied Probability 311–337. Chapman and Hall.
  • Papaspiliopoulos, O., Roberts, G. O. and Stramer, O. (2013). Data Augmentation for Diffusions., J. Comput. Graph. Statist. 22 665–688.
  • Pedersen, A. R. (1995). Consistency and asymptotic normality of an approximate maximum likelihood estimator for discretely observed diffusion processes., Bernoulli 1 257–279.
  • Roberts, G. O. and Stramer, O. (2001). On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm., Biometrika 88 603–621.
  • Rogers, L. C. G. and Williams, D. (2000)., Diffusions, Markov processes, and martingales. Vol. 2. Cambridge Mathematical Library. Cambridge University Press, Cambridge. Itô calculus, Reprint of the second (1994) edition.
  • Rosenthal, J. S. (2011)., Handbook of Markov Chain Monte Carlo (Chapman & Hall/CRC Handbooks of Modern Statistical Methods), 1 ed. Optimal proposal distributions and adaptive MCMC. Chapman and Hall/CRC.
  • Schauer, M. R., Van der Meulen, F. H. and Van Zanten, J. H. (2017). Guided proposals for simulating multi-dimensional diffusion bridges., Bernoulli 23 2917–2950.
  • Sermaidis, G., Papaspiliopoulos, O., Roberts, G. O., Beskos, A. and Fearnhead, P. (2013). Markov chain Monte Carlo for exact inference for diffusions., Scand. J. Stat. 40 294–321.
  • Sørensen, H. (2004). Parametric Inference for Diffusion Processes Observed at Discrete Points in Time: a Survey., Internat. Statist. Rev. 72 337–354.
  • Steiner, A. and Gander, M. J. (1999). Parametrische Lösungen der Räuber-Beute-Gleichungen im Vergleich., Il Volterriano 7 32–44.
  • Stramer, O. and Bognar, M. (2011). Bayesian inference for irreducible diffusion processes using the pseudo-marginal approach., Bayesian Anal. 6 231–258.
  • Tierney, L. (1998). A note on Metropolis-Hastings kernels for general state spaces., Ann. Appl. Probab. 8 1–9.
  • Van der Meulen, F. H., Schauer, M. and Van Zanten, J. H. (2014). Reversible jump MCMC for nonparametric drift estimation for diffusion processes., Comput. Statist. Data Anal. 71 615–632.
  • Van der Meulen, F. H. and Schauer, M. R. (2016). Bayesian estimation of incompletely observed diffusions., ArXiv e-prints.
  • Van der Meulen, F. H. and Van Zanten, J. H. (2013). Consistent nonparametric Bayesian inference for discretely observed scalar diffusions., Bernoulli 19 44–63.
  • Van Zanten, J. H. (2013). Nonparametric Bayesian methods for one-dimensional diffusion models., Mathematical biosciences 243 215–222.
  • Vats, D., Flegal, J. M. and Jones, G. L. (2015). Multivariate Output Analysis for Markov chain Monte Carlo., ArXiv e-prints.
  • Whitaker, G. A., Golightly, A., Boys, R. J. and Sherlock, C. (2017). Improved bridge constructs for stochastic differential equations., Statistics and Computing 27 885–900.