Consistent nonparametric Bayesian inference for discretely observed scalar diffusions. (English) Zbl 1259.62070

Summary: We study Bayes procedures for the problem of nonparametric drift estimation for one-dimensional, ergodic diffusion models from discrete-time, low-frequency data. We give conditions for posterior consistency and verify these conditions for concrete priors, including priors based on wavelet expansions.


62M05 Markov processes: estimation; hidden Markov models
62G05 Nonparametric estimation
62F15 Bayesian inference
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
Full Text: DOI arXiv Euclid


[1] Barron, A., Schervish, M.J. and Wasserman, L. (1999). The consistency of posterior distributions in nonparametric problems. Ann. Statist. 27 536-561. · Zbl 0980.62039
[2] 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. · Zbl 1100.62079
[3] Birgé, L. (1982). Tests robustes pour des variables indépendantes et des chaînes de Markov. Ann. Sci. Univ. Clermont-Ferrand II Math. 20 70-77. · Zbl 0554.62031
[4] Borodin, A.N. and Salminen, P. (2002). Handbook of Brownian Motion-Facts and Formulae , 2nd ed. Probability and Its Applications . Basel: Birkhäuser. · Zbl 1012.60003
[5] Chib, S., Shephard, N. and Pitt, M. (2010). Likelihood based inference for diffusion driven state space models.
[6] Daubechies, I. (1992). Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics 61 . Philadelphia, PA: SIAM. · Zbl 0776.42018
[7] Diaconis, P. and Freedman, D. (1986). On the consistency of Bayes estimates. Ann. Statist. 14 1-26. · Zbl 0595.62022
[8] Elerian, O., Chib, S. and Shephard, N. (2001). Likelihood inference for discretely observed nonlinear diffusions. Econometrica 69 959-993. · Zbl 1017.62068
[9] Eraker, B. (2001). MCMC analysis of diffusion models with application to finance. J. Bus. Econom. Statist. 19 177-191.
[10] Ghosal, S., Ghosh, J.K. and van der Vaart, A.W. (2000). Convergence rates of posterior distributions. Ann. Statist. 28 500-531. · Zbl 1105.62315
[11] Ghosal, S. and Tang, Y. (2006). Bayesian consistency for Markov processes. Sankhyā 68 227-239. · Zbl 1193.62035
[12] Ghosal, S. and van der Vaart, A. (2007). Convergence rates of posterior distributions for non-i.i.d. observations. Ann. Statist. 35 192-223. · Zbl 1114.62060
[13] Golightly, A. and Wilkinson, D.J. (2008). Bayesian inference for nonlinear multivariate diffusion models observed with error. Comput. Statist. Data Anal. 52 1674-1693. · Zbl 1452.62603
[14] Härdle, W., Kerkyacharian, G., Picard, D. and Tsybakov, A. (1998). Wavelets , Approximation , and Statistical Applications. Lecture Notes in Statistics 129 . New York: Springer. · Zbl 0899.62002
[15] Hernández, E. and Weiss, G. (1996). A First Course on Wavelets. Studies in Advanced Mathematics . Boca Raton, FL: CRC Press. With a foreword by Yves Meyer. · Zbl 0885.42018
[16] Itô, K. and McKean, H.P. Jr. (1965). Diffusion Processes and Their Sample Paths . Berlin: Springer. · Zbl 0127.09503
[17] Jensen, B. and Poulsen, R. (2002). Transition densities of diffusion processes: Numerical comparison of approximation techniques. Journal of Derivatives 9 1-15.
[18] Kallenberg, O. (1997). Foundations of Modern Probability. Probability and Its Applications ( New York ). New York: Springer. · Zbl 0892.60001
[19] Karatzas, I. and Shreve, S.E. (1991). Brownian Motion and Stochastic Calculus , 2nd ed. Graduate Texts in Mathematics 113 . New York: Springer. · Zbl 0734.60060
[20] Roberts, G.O. and Stramer, O. (2001). On inference for partially observed nonlinear diffusion models using the Metropolis-Hastings algorithm. Biometrika 88 603-621. · Zbl 0985.62066
[21] Rogers, L.C.G. and Williams, D. (1987). Diffusions , Markov Processes , and Martingales. Vol. 2, Itô Calculus. Wiley Series in Probability and Mathematical Statistics : Probability and Mathematical Statistics . New York: Wiley. · Zbl 0977.60005
[22] Schwartz, L. (1965). On Bayes procedures. Z. Wahrsch. Verw. Gebiete 4 10-26. · Zbl 0158.17606
[23] Shen, X. and Wasserman, L. (2001). Rates of convergence of posterior distributions. Ann. Statist. 29 687-714. · Zbl 1041.62022
[24] Tang, Y. and Ghosal, S. (2007). Posterior consistency of Dirichlet mixtures for estimating a transition density. J. Statist. Plann. Inference 137 1711-1726. · Zbl 1118.62089
[25] Walker, S. (2004). New approaches to Bayesian consistency. Ann. Statist. 32 2028-2043. · Zbl 1056.62040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.