Skip to main content
Log in

Asymptotic Behaviour of Trajectory Fitting Estimators for Certain Non-ergodic SDE

  • Published:
Statistical Inference for Stochastic Processes Aims and scope Submit manuscript

Abstract

Suppose one observes a path of a stochastic processX = (Xt)t≥0 driven by the equation

dXt=θ a(Xt)dt + dWt, t≥0, θ ≥ 0

with a(x) = x or a(x) = |x|α for some α ∈ [0,1) and given initial condition X 0. If the true but unknown parameter θ0 is positive then X is non-ergodic. It is shown that in this situation a trajectory fitting estimator for θ0 is strongly consistent and has the same limiting distribution as the maximum likelihood estimator, but converges of minor order.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Basawa, I. V. and Prakasa Rao, B. L. S.: Statistical Inference for Stochastic Processes, Academic Press, London New York et al., 1980.

    MATH  Google Scholar 

  2. Basawa, I. V. and Scott, D. J.: Asymptotic Optimal Inference for Non-ergodic Models, Lecture Notes in Statistics 17, Springer-Verlag, New York, Heidelberg, Berlin, 1983.

    MATH  Google Scholar 

  3. Dietz, H. M. and Kutoyants, Y.: A minimum distance estimator for diffusion processes with ergodic properties, Technical Report 17, Institute of Applied Analysis and Stochastics. Berlin, 1992.

  4. Dietz, H. M. and Kutoyants, Y.: A class of minimum-distance estimators for diffusion processes with ergodic properties, Statist. & Decisions, 15 (1997), 211-227.

    MATH  MathSciNet  Google Scholar 

  5. Feigin, P. D.: Some comments concerning curious singularity, J. Appl. Prob. 16 (1979) 440-444.

    Article  MATH  MathSciNet  Google Scholar 

  6. Gichman, I. I. and Skorochod, A.W.: Stochastische Differentialgleichungen, Akademie Verlag, Berlin, 1972.

    MATH  Google Scholar 

  7. Kutoyants, Y.: On a property of the parameter estimator of the trend coefficient (in russian), Izvestiya Akademii Nauk Arm. SSR, Mathematika, 12(4) (1977), 245-251.

    MATH  Google Scholar 

  8. Kutoyants, Y.: Minimum distance parameter estimation for diffusion type observations. C.R. Acad. Sci. Paris, Serie I 312 (1991), 637-642.

    MATH  MathSciNet  Google Scholar 

  9. Kutoyants, Y.: Statistical Inference for Ergodic Diffusion Processes.-, forthcoming.

  10. Liptser, R. S. and Shiryaev, A. N.: Statistics of Random Processes II. Applications, Springer-Verlag, New York, Heidelberg, Berlin, 1978.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dietz, H.M. Asymptotic Behaviour of Trajectory Fitting Estimators for Certain Non-ergodic SDE. Statistical Inference for Stochastic Processes 4, 249–258 (2001). https://doi.org/10.1023/A:1012254332474

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1012254332474

Navigation