Journal of Statistical Physics

, Volume 144, Issue 1, pp 150–170 | Cite as

Parametric Estimation of Stationary Stochastic Processes Under Indirect Observability

  • R. Azencott
  • A. Beri
  • I. Timofeyev


For many natural turbulent dynamic systems, observed high dimensional dynamic data can be approximated at slow time scales by a process X t driven by a systems of stochastic differential equations (SDEs). When one tries to estimate the parameters of this unobservable SDEs systems, there is a clear mismatch between the available data and the SDEs dynamics to be parametrized. Here, we formalize this Indirect Observability framework as follows.

We consider an unobservable centered stationary Gaussian process X t with covariance function K(u,θ)=E[X t X t+u ], parametrized by an unknown vector θ which lies in a compact subset Θ of ℝ p . We assume that the only observable data are generated by centered stationary processes \(Y_{t}^{\varepsilon }\), indexed by a scale separation parameter ε>0. These approximating processes have arbitrary probability distributions, exponentially decaying covariances, and are assumed to converge to X t in L 4 as ε→0. We show how to construct estimators of the underlying parameter vector θ which depend only on the observable data \(Y_{t}^{\varepsilon }\), and converge to the true parameter values as ε→0.

We study adaptive subsampling schemes involving [N(ε)+k(ε)]→∞ observations \(V_{n} = Y^{\varepsilon }_{n \Delta(\varepsilon )}\) extracted from the approximating process \(Y^{\varepsilon }_{t}\) by subsampling at time intervals Δ(ε)→0. We focus on parameter estimators which are smooth functions of subsampled empirical covariance estimators \(\hat{r}_{k}(N,\Delta)\) associated to non vanishing time lags k(ε)Δ(ε) tending to fixed positive limits as ε→0.

We show that provided lim  ε→0 N(ε)Δ(ε)=+∞, these subsampled approximate covariance estimators converge in L 2 to the true covariance function K(u,θ) of X t for all u,θ. Applying a generic version of the method of moments suitably boosted up by adequately adjusted multiple subsampling schemes, we show that this implies, in a very wide range of situations, the existence of consistent estimators \(\hat{\theta}(\varepsilon )\) of the unknown parameter vector θ, based only on adequately subsampled approximate data \(Y^{\varepsilon }_{t}\).


Gaussian processes Empirical covariance estimators Adaptive sub-sampling Indirect observability Non vanishing lags 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ait-Sahalia, Y., Mykland, P., Zhang, L.: How often to sample a continuous-time process in the presence of market microstructure noise. Rev. Financ. Stud. 18(2), 351 (2005) CrossRefGoogle Scholar
  2. 2.
    Arnold, L., Imkeller, P., Wu, Y.: Reduction of deterministic coupled atmosphere–ocean models to stochastic ocean models: a numerical case study of the Lorenz–Maas system. Dyn. Syst. 18(4), 295–350 (2003) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Azencott, R., Dacunha-Castelle, D.: Series of Irregular Observations: Forecasting and Model Building. Springer, Berlin (1986) MATHCrossRefGoogle Scholar
  4. 4.
    Azencott, R., Beri, A., Timofeyev, I.: Adaptive sub-sampling for parametric estimation of Gaussian diffusions. J. Stat. Phys. 139(6), 1066–1089 (2010) MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    Azencott, R., Beri, A., Timofeyev, I.: Sub-sampling in parametric estimation of stochastic differential equations from discrete data (2010, submitted) Google Scholar
  6. 6.
    Barndorff-Nielsen, O., Shephard, N.: Estimating quadratic variation using realized variance. J. Appl. Econom. 17(5), 457–477 (2002) CrossRefGoogle Scholar
  7. 7.
    Berner, J.: Linking nonlinearity and non-Gaussianity of planetary wave behavior by the Fokker–Planck equation. J. Atmos. Sci. 62, 2098–2117 (2005) MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Crommelin, D., Vanden-Eijnden, E.: Diffusion estimation from multiscale data by operator eigenpairs (2010, submitted) Google Scholar
  9. 9.
    Culina, J., Kravtsov, S., Monahan, A.H.: Stochastic parameterisation schemes for use in realistic climate models J. Atmos. Sci. 68, 284–299 (2010) ADSCrossRefGoogle Scholar
  10. 10.
    DelSole, T.: A fundamental limitation of Markov models. J. Atmos. Sci. 57, 2158–2168 (2000) ADSCrossRefGoogle Scholar
  11. 11.
    Deuflhard, P., Schütte, C.: Molecular conformation dynamics and computational drug design. In: Applied Mathematics Entering the 21st Century: Invited Talks from the ICIAM 2003 Congress, p. 91. Society for Industrial Mathematics, Philadelphia (2004) Google Scholar
  12. 12.
    Franzke, C., Majda, A.J.: Low-order stochastic mode reduction for a prototype atmospheric GCM. J. Atmos. Sci. 63, 457–479 (2006) MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Franzke, C., Majda, A.J., Vanden-Eijnden, E.: Low-order stochastic mode reduction for a realistic barotropic model climate. J. Atmos. Sci. 62, 1722–1745 (2005) MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    Hasselman, K.: Stochastic climate models. Part I: Theory. Tellus 28, 473–485 (1976) ADSCrossRefGoogle Scholar
  15. 15.
    Heston, S.: A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud. 6(2), 327 (1993) CrossRefGoogle Scholar
  16. 16.
    Hummer, G.: Position-dependent diffusion coefficients and free energies from Bayesian analysis of equilibrium and replica molecular dynamics simulations. New J. Phys. 7, 34 (2005) CrossRefGoogle Scholar
  17. 17.
    Majda, A.J., Timofeyev, I., Vanden-Eijnden, E.: A priori tests of a stochastic mode reduction strategy. Physica D 170, 206–252 (2002) MathSciNetADSMATHCrossRefGoogle Scholar
  18. 18.
    Majda, A.J., Timofeyev, I., Vanden-Eijnden, E.: Systematic strategies for stochastic mode reduction in climate. J. Atmos. Sci. 60(14), 1705–1722 (2003) MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    Papavasiliou, A., Pavliotis, G.A., Stuart, A.: Maximum likelihood drift estimation for multiscale diffusions. Stoch. Process. Appl. 119(10), 3173–3210 (2009) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Pavliotis, G.A., Stuart, A.: Parameter estimation for multiscale diffusions. J. Stat. Phys. 127, 741–781 (2007) MathSciNetADSMATHCrossRefGoogle Scholar
  21. 21.
    Zhang, L., Mykland, P., Ait-Sahalia, Y.: A tale of two time scales. J. Am. Stat. Assoc. 100(472), 1394–1411 (2005) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA
  2. 2.Ecole Normale SuperieureParisFrance

Personalised recommendations