Electronic Journal of Statistics

The nonparametric LAN expansion for discretely observed diffusions

Sven Wang

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Consider a scalar reflected diffusion $(X_{t}:t\geq 0)$, where the unknown drift function $b$ is modelled nonparametrically. We show that in the low frequency sampling case, when the sample consists of $(X_{0},X_{\Delta },...,X_{n\Delta })$ for some fixed sampling distance $\Delta >0$, the model satisfies the local asymptotic normality (LAN) property, assuming that $b$ satisfies some mild regularity assumptions. This is established by using the connections of diffusion processes to elliptic and parabolic PDEs. The key tools used are regularity estimates for certain parabolic PDEs as well as a detailed analysis of the spectral properties of the elliptic differential operator related to $(X_{t}:t\geq 0)$.

Article information

Electron. J. Statist., Volume 13, Number 1 (2019), 1329-1358.

Received: March 2018
First available in Project Euclid: 5 April 2019

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Zentralblatt MATH identifier

Primary: 62M99: None of the above, but in this section

Nonparametric diffusion model LAN property parabolic PDE

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Wang, Sven. The nonparametric LAN expansion for discretely observed diffusions. Electron. J. Statist. 13 (2019), no. 1, 1329--1358. doi:10.1214/19-EJS1545. https://projecteuclid.org/euclid.ejs/1554451244

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  • [1] K. Abraham. Nonparametric Bayesian posterior contraction rates for scalar diffusions with high-frequency data., Bernoulli, to appear, arXiv :1802.05635, 2018.
  • [2] D. Bakry, I. Gentil, and M. Ledoux., Analysis and Geometry of Markov Diffusion Operators. Springer, 2013.
  • [3] R. Bass., Diffusions and Elliptic Operators. Springer, 1998.
  • [4] B. Brown. Martingale Central Limit Theorems., Ann. Math. Statist., 42:59–66, 1971.
  • [5] I. Castillo and R. Nickl. Nonparametric Bernstein-von Mises theorems in Gaussian White Noise., Annals of Statistics, 41 :1999–2028, 2013.
  • [6] I. Castillo and R. Nickl. On the Bernstein - von Mises phenomenon for nonparametric Bayes procedures., Annals of Statistics, 42 :1941–1969, 2014.
  • [7] J. Chorowski., Statistics for diffusion processes with low and high-frequency observations. PhD Thesis, 2016.
  • [8] E. B. Davies., Spectral Theory and Differential Operators. Cambridge University Press, 1995.
  • [9] I. Gihman and A. Skorohod., Stochastic differential equations. Springer, 1972.
  • [10] E. Giné and R. Nickl., Mathematical Foundations of Infinite-Dimensional statistical models. Cambridge University Press, 2016.
  • [11] E. Gobet. LAN property for ergodic diffusions with discrete observations., Annales de l’Institut Henri Poincaré B, 38:711–737, 2002.
  • [12] E. Gobet, M. Hoffmann, and M. Reiß. Nonparametric estimation of scalar diffusions based on low frequency data., Annals of Statistics, 32, 2004.
  • [13] S. Gugushvili and P. Spreij. Nonparametric Bayesian drift estimation for multidimensional stochastic differential equations., Lith. Math. J., 54:127–141, 2014.
  • [14] L. P. Hansen, J. A. Scheinkman, and N. Touzi. Spectral methods for identifying scalar diffusions., Journal of Econometrics, 86:1–32, 1998.
  • [15] M. Hoffmann, A. Munk, and J. Schmidt-Hieber. Adaptive wavelet estimation of the diffusion coefficient under additive error measurements., Annals of the Institute H. Poincaré, 48 :1186–1216, 2012.
  • [16] J. Koskela, D. Spano, and P. A. Jenkins. Consistency of Bayesian nonparametric inference for discretely observed jump diffusions., arXiv :1506.04709, 2015.
  • [17] J. Lions and E. Magenes., Non-Homogeneous Boundary Value Problems and Applications, Vol. 1. Springer, 1972.
  • [18] Y. Lu. On the bernstein-von mises theorem for high dimensional nonlinear bayesian inverse problems., arXiv :1706.00289, 2017.
  • [19] A. Lunardi., Analytic Semigroups and Optimal Regularity for Parabolic Problems. Birkhäuser, 1995.
  • [20] F. Monard, R. Nickl, and G. Paternain. Efficient nonparametric Bayesian inference for X-ray transforms., Annals of Statistics, to appear.
  • [21] R. Nickl. Bernstein - von Mises theorems for statistical inverse problems I: Schrödinger equation., Journal of the European Mathematical Society, to appear.
  • [22] R. Nickl and K. Ray. Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions., arXiv :1810.01702, 2018.
  • [23] R. Nickl and J. Söhl. Bernstein - von Mises theorems for statistical inverse problems II: compound Poisson processes., arXiv preprint, 2017.
  • [24] R. Nickl and J. Söhl. Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions., Annals of Statistics, 45 :1664–1693, 2017.
  • [25] O. Papaspiliopoulos, Y. Pokern, G. O. Roberts, and A. M. Stuart. Nonparametric estimation of diffusions: a differential equations approach., Biometrika, 99:511–531, 2012.
  • [26] F. van der Meulen, M. Schauer, and H. van Zanten. Reversible jump MCMC for nonparametric drift estimation for diffusion processes., Comput. Statist. Data Anal., 71:615–632, 2014.
  • [27] A. W. van der Vaart., Asymptotic Statistics. Cambridge University Press, 1998.
  • [28] J. van Waaij and H. van Zanten. Gaussian process methods for one-dimensional diffusions: Optimal rates and adaptation., Electron. J. Stat., 10:628–645, 2013.