Trajectory fitting estimators for SPDEs driven by additive noise

  • Igor CialencoEmail author
  • Ruoting Gong
  • Yicong Huang


In this paper we study the problem of estimating the drift/viscosity coefficient for a large class of linear, parabolic stochastic partial differential equations (SPDEs) driven by an additive space-time noise. We propose a new class of estimators, called trajectory fitting estimators (TFEs). The estimators are constructed by fitting the observed trajectory with an artificial one, and can be viewed as an analog to the classical least squares estimators from the time-series analysis. As in the existing literature on statistical inference for SPDEs, we take a spectral approach, and assume that we observe the first N Fourier modes of the solution, and we study the consistency and the asymptotic normality of the TFE, as \(N\rightarrow \infty \).


Stochastic partial differential equations Trajectory fitting estimator Parameter estimation Inverse problems Estimation of viscosity coefficient 

Mathematics Subject Classification

60H15 35Q30 65L09 



Part of the research was performed while Igor Cialenco was visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation. The authors would like to thank the anonymous referees, the associate editor and the editor for their helpful comments and suggestions which improved greatly the final manuscript.


  1. Bishwal JPN (2008) Parameter estimation in stochastic differential equations. Lecture notes in mathematics, vol 1923. Springer, BerlinGoogle Scholar
  2. Chow P (2007) Stochastic partial differential equations. Chapman & Hall/CRC applied mathematics and nonlinear science series. Chapman & Hall/CRC, Boca RatonGoogle Scholar
  3. Cialenco I, Glatt-Holtz N (2011) Parameter estimation for the stochastically perturbed Navier-Stokes equations. Stoch Process Appl 121(4):701–724MathSciNetCrossRefzbMATHGoogle Scholar
  4. Cialenco I, Lototsky SV (2009) Parameter estimation in diagonalizable bilinear stochastic parabolic equations. Stat Inference Stoch Process 12(3):203–219MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cialenco I, Xu L (2014) A note on error estimation for hypothesis testing problems for some linear SPDEs. Stoch Partial Differ Equ Anal Comput 2(3):408–431MathSciNetzbMATHGoogle Scholar
  6. Cialenco I, Xu L (2015) Hypothesis testing for stochastic PDEs driven by additive noise. Stoch Process Appl 125(3):819–866MathSciNetCrossRefzbMATHGoogle Scholar
  7. Huebner M, Rozovskii BL (1995) On asymptotic properties of maximum likelihood estimators for parabolic stochastic PDE’s. Probab Theory Relat Fields 103(2):143–163MathSciNetCrossRefzbMATHGoogle Scholar
  8. Huebner M, Lototsky SV, Rozovskii BL (1997) Asymptotic properties of an approximate maximum likelihood estimator for stochastic PDEs. In: Statistics and control of stochastic processes (Moscow, 1995/1996). World Scientific Publishing, Singapore, pp 139–155Google Scholar
  9. Kutoyants YA (2004) Statistical inference for ergodic diffusion processes. Springer series in statistics. Springer, LondonGoogle Scholar
  10. Kutoyants YA (1991) Minimum-distance parameter estimation for diffusion-type observations. C R Acad Sci Paris Sér I Math 312(8):637–642MathSciNetzbMATHGoogle Scholar
  11. Lototsky SV (2009) Statistical inference for stochastic parabolic equations: a spectral approach. Publ Mat 53(1):3–45MathSciNetCrossRefzbMATHGoogle Scholar
  12. Markussen B (2003) Likelihood inference for a discretely observed stochastic partial differential equation. Bernoulli 9(5):745–762MathSciNetCrossRefzbMATHGoogle Scholar
  13. Mishra MN, Bishwal JPN (1995) Approximate maximum likelihood estimation for diffusion processes from discrete observations. Stoch Stoch Rep 52(1–2):1–13MathSciNetCrossRefzbMATHGoogle Scholar
  14. Mishra MN, Prakasa Rao BLS (2002) Approximation of maximum likelihood estimator for diffusion processes from discrete observations. Stoch Anal Appl 20(6):1309–1329MathSciNetCrossRefzbMATHGoogle Scholar
  15. Piterbarg LI, Rozovskii BL (1997) On asymptotic problems of parameter estimation in stochastic PDE’s: discrete time sampling. Math Methods Stat 6(2):200–223MathSciNetzbMATHGoogle Scholar
  16. Prakasa Rao BLS (2003) Estimation for some stochastic partial differential equations based on discrete observations. II. Calcutta Stat Assoc Bull 54(215–216):129–141MathSciNetzbMATHGoogle Scholar
  17. Rozovskii BL (1990) Stochastic evolution systems. Mathematics and its applications (soviet series), vol 35. Kluwer Academic Publishers Group, Dordrecht. Linear theory and applications to nonlinear filteringGoogle Scholar
  18. Shiryaev AN (1996) Probability. Graduate texts in mathematics, vol 95, 2nd edn. Springer, New YorkGoogle Scholar
  19. Shubin MA (2001) Pseudodifferential operators and spectral theory, 2nd edn. Springer, BerlinCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA

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