Statistical Inference for Stochastic Processes

, Volume 13, Issue 3, pp 175–192 | Cite as

Drift estimation for a periodic mean reversion process

  • Herold DehlingEmail author
  • Brice Franke
  • Thomas Kott


In this paper we propose a periodic, mean-reverting Ornstein–Uhlenbeck process of the form
$$ dX_t=(L(t)-\alpha\, X_t)\, dt + \sigma\, dB_t, \quad t\geq 0, $$
where L(t) is a periodic, parametric function. We apply maximum likelihood estimation for the drift parameters based on time-continuous observations. The estimator is given explicitly and we prove strong consistency and asymptotic normality as the observed number of periods tends to infinity. The essential idea of the asymptotic study is the interpretation of the stochastic process as a sequence of random variables that take values in some function space.


Time-inhomogeneous diffusion process Ornstein–Uhlenbeck process Maximum likelihood estimation Asymptotic normality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bishwal JPN (2008) Parameter estimation in stochastic differential equations. Springer-Verlag, BerlinzbMATHCrossRefGoogle Scholar
  2. Gantmacher FR (1986) Matrizentheorie. VEB Deutscher Verlag der Wissenschaften, BerlinzbMATHGoogle Scholar
  3. Geman H (2005) Commodities and commodity derivatives. Wiley, ChichesterGoogle Scholar
  4. Kuo HH (2006) Introduction to stochastic integrals. Springer-Verlag, New YorkGoogle Scholar
  5. Kutoyants YA (2004) Statistical inference for ergodic diffusion processes. Springer-Verlag, LondonzbMATHGoogle Scholar
  6. Lax PD (2002) Functional analysis. Wiley, New YorkzbMATHGoogle Scholar
  7. Lipster RS, Shiryayev AN (1977) Statistics of random processes I. Springer-Verlag, BerlinGoogle Scholar
  8. Øksendal B (2003) Stochastic Differential Equations. Springer-Verlag, BerlinGoogle Scholar
  9. Ornstein LS, Uhlenbeck GE (1930) On the theory of Brownian motion. Phys Rev 36: 823–841CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany

Personalised recommendations