Bayesian Analysis

Comment on Article by Chkrebtii, Campbell, Calderhead, and Girolami

Martin Lysy

Full-text: Open access

Abstract

The authors present an ingenious probabilistic numerical solver for deterministic differential equations (DEs). The true solution is progressively identified via model interrogations, in a formal framework of Bayesian updating. I have attempted to extend the authors’ ideas to stochastic differential equations (SDEs), and discuss two challenges encountered in this endeavor: (i) the non-differentiability of SDE sample paths, and (ii) the sampling of diffusion bridges, typically required of solutions to the SDE inverse problem.

Article information

Source
Bayesian Anal., Volume 11, Number 4 (2016), 1269-1273.

Dates
First available in Project Euclid: 30 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.ba/1480474948

Digital Object Identifier
doi:10.1214/16-BA1036

Mathematical Reviews number (MathSciNet)
MR3577379

Zentralblatt MATH identifier
1357.62119

Keywords
stochastic differential equations probabilistic solution diffusion bridge sampling

Citation

Lysy, Martin. Comment on Article by Chkrebtii, Campbell, Calderhead, and Girolami. Bayesian Anal. 11 (2016), no. 4, 1269--1273. doi:10.1214/16-BA1036. https://projecteuclid.org/euclid.ba/1480474948


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References

  • Andrieu, C., Doucet, A., and Holenstein, R. (2010). “Particle Markov chain Monte Carlo Methods.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 72: 1–33.
  • Bladt, M., Finch, S., and Sørensen, M. (2016). “Simulation of multivariate diffusion bridges.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 78(2): 343–369.
  • Chkrebtii, O. A., Campbell, D. A., Calderhead, B., and Girolami, M. A. (2016). “Bayesian solution uncertainty quantification for differential equations.” Bayesian Analysis.
  • Lysy, M. and Pillai, N. S. (2013). “Statistical Inference for Stochastic Differential Equations with Memory.” Technical report, University of Waterloo.
  • Roberts, G. O. and Stramer, O. (2001). “On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm.” Biometrika, 88(3): 603–621.
  • Van Kampen, N. G. (1981). “Itô versus Stratonovich.” Journal of Statistical Physics, 24(1): 175–187.

See also

  • Related item: Oksana A. Chkrebtii, David A. Campbell, Ben Calderhead, Mark A. Girolami (2016). Bayesian Solution Uncertainty Quantification for Differential Equations. Bayesian Anal. Vol. 11, Iss. 4, 1239–1267.