Inference for a change-point problem under a generalised Ornstein–Uhlenbeck setting

  • Fuqi Chen
  • Rogemar MamonEmail author
  • Sévérien Nkurunziza


Determining accurately when regime and structural changes occur in various time-series data is critical in many social and natural sciences. We develop and show further the equivalence of two consistent estimation techniques in locating the change point under the framework of a generalised version of the one-dimensional Ornstein–Uhlenbeck process. Our methods are based on the least sum of squared error and the maximum log-likelihood approaches. The case where both the existence and the location of the change point are unknown is investigated and an informational methodology is employed to address these issues. Numerical illustrations are presented to assess the methods’ performance.


Sequential analysis Least sum of squared errors Maximum likelihood Consistent estimator Existence of change point 



F. Chen and R. Mamon would like to acknowledge the financial support, via a Fields Postdoctoral Fellowship for F. Chen, provided by the Fields Institute for Research in Mathematical Sciences, Toronto, Ontario, Canada, where this research was initially conceived and partially conducted. R. Mamon wishes to express his appreciation for the hospitality of the Division of Physical Sciences and Mathematics, University of the Philippines Visayas, where certain revisions of this paper were made during his academic visit both as an Adjunct Professor and a DOST-PCIEERD Balik Scientist for the Philippine government. Both S. Nkurunziza and R. Mamon thank the Natural Sciences and Engineering Research Council of Canada for the support of their research. Likewise, all authors are grateful to Charmaine Dean (Western University’s Dean of Science) and Matt Davison (Canada Research Chair in Quantitative Finance) for their generous financial support on this research collaboration. Some valuable feedback from Charmaine Dean and Reg Kuperger are very much appreciated. Finally, the authors sincerely thank the Associate Editor and an anonymous referee for their helpful comments.


  1. Aalen, O., Gjessing, H. (2004). Survival models based on the Ornstein-Uhlenbeck process. Lifetime Data Analysis, 10, 407–423.Google Scholar
  2. Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In B. Petrov, F. Csáki (Eds.), 2nd international symposium on information theory, Tsahkadsor, Armenia, USSR, September 2–8, 1971, Budapest: Akadémiai Kiadó, pp. 267–281.Google Scholar
  3. Auger, I., Lawrence, C. (1989). Algorithms for the optimal identification of segment neighborhoods. Bulletin of Mathematical Biology, 51, 39–54.Google Scholar
  4. Bai, J., Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica, 66, 47–78.Google Scholar
  5. Benth, F., Koekebakker, S., Taib, C. (2015). Stochastic dynamical modelling of spot freight rates. IMA Journal of Management Mathematics, 26, 273–297.Google Scholar
  6. Chen, S. (2010). Modelling the dynamics of commodity prices for investment decisions under uncertainty. Ph.D. dissertation, University of Waterloo, Canada.Google Scholar
  7. Chen, F., Nkurunziza, S. (2015). Optimal method in multiple regression with structural changes. Bernoulli, 21, 2217–2241.Google Scholar
  8. Date, P., Bustreo, R. (2016). Value-at-risk for fixed-income portfolios: A Kalman filtering approach. IMA Journal of Management Mathematics, 27, 557–573.Google Scholar
  9. Date, P., Mamon, R., Tenyakov, A. (2013). Filtering and forecasting commodity futures prices under an HMM framework. Energy Economics, 40, 1001–1013.Google Scholar
  10. De Gregorio, A., Iacus, S. (2008). Least squares volatility change point estimation for partially observed diffusion processes. Communications in Statistics: Theory and Methods, 37, 2342–2357.Google Scholar
  11. Dehling, H., Franke, B., Kott, T. (2010). Drift estimation for a periodic mean reversion process. Statistical Inference for Stochastic Processes, 13, 175–192.Google Scholar
  12. Dehling, H., Franke, B., Kott, T., Kulperger, R. (2014). Change point testing for the drift parameters of a periodic mean reversion process. Statistical Inference for Stochastic Process, 17, 1–18.Google Scholar
  13. Elias, R., Wahab, M., Fung, F. (2014). A comparison of regime-switching temperature modeling approaches for applications in weather derivatives. European Journal of Operational Research, 232, 549–560.Google Scholar
  14. Elliott, R., Wilson, C. (2007). The term structure of interest rates in a hidden Markov setting. In R. Mamon, R. Elliott (Eds.), Hidden Markov models in finance (pp. 14–31). New York, NY: Springer.Google Scholar
  15. Erlwein, C., Benth, F., Mamon, R. (2010). HMM filtering and parameter estimation of an electricity spot price model. Energy Economics, 32, 1034–1043.Google Scholar
  16. Gallagher, C., Lund, R., Robbins, M. (2012). Changepoint detection in daily precipitation data. Environmetrics, 23, 407–419.Google Scholar
  17. Gombay, E. (2010). Change detection in linear regression with time series errors. Canadian Journal of Statistics, 38, 65–79.MathSciNetzbMATHGoogle Scholar
  18. Howell, S., Duck, P., Hazel, A., Johnson, P., Pinto, H., Strbac, G., et al. (2011). A partial differential equation system for modelling stochastic storage in physical systems with applications to wind power generation. IMA Journal of Management Mathematics, 22, 231–252.MathSciNetCrossRefzbMATHGoogle Scholar
  19. Iacus, S., Yoshida, N. (2012). Estimation for the change point of volatility in a stochastic differential equation. Stochastic Processes and Applications, 122, 1068–1092.Google Scholar
  20. Killick, R., Fearnhead, P., Eckley, I. (2012). Optimal detection of change points with a linear computational cost. Journal of the American Statistical Association, 107, 1590–1598.Google Scholar
  21. Lánský, P., Sacerdote, L. (2001). The Ornstein-Uhlenbeck neuronal model with signal-dependent noise. Physics Letters A, 285, 132–140.Google Scholar
  22. Le Breton, A. (1976). On continuous and discrete sampling for parameter estimation in diffusion type processes. Mathematical Programming Studies, 5, 124–144.MathSciNetCrossRefGoogle Scholar
  23. Lee, S. (2011). Change point test for dispersion parameter based on discretely observed sample from SDE models. Bulletin of the Korean Mathematical Society, 48, 839–845.MathSciNetCrossRefzbMATHGoogle Scholar
  24. Lee, A., Guo, M. (2015). Monitoring change point for diffusion parameter based on discretely observed sample from stochastic differential equation models. Applied Stochastic Models in Business and Industry, 31, 609–625.Google Scholar
  25. Liang, Z., Yuen, K., Guo, J. (2011). Optimal proportional reinsurance and investment in a stock market with Ornstein-Uhlenbeck process. Insurance: Mathematics and Economics, 49, 207–215.Google Scholar
  26. Lipster, R., Shiryaev, A. (2001). Statistics of random processes I. Berlin: Springer.Google Scholar
  27. Lu, Q., Lund, R. (2007). Simple linear regression with multiple level shifts. Canadian Journal of Statistics, 35, 447–458.Google Scholar
  28. Lu, S. (2003). Ornstein-Uhlenbeck diffusion quantum Monte Carlo calculations for small first-row polyatomic molecules. Journal of Chemical Physics, 118, 9528–9532.CrossRefGoogle Scholar
  29. Lu, S. (2004). Ornstein-Uhlenbeck diffusion quantum Monte Carlo study on the bond lengths and harmonic frequencies of some first-row diatomic molecules. Journal of Chemical Physics, 120. doi: 10.1063/1.1639370.
  30. Nkurunziza, S., Zhang, P. (2016). Estimation and testing in generalized mean-reverting processes with change-point. Statistical Inference for Stochastic Processes. to appear. doi: 10.1007/s11203-016-9151-3.
  31. Page, E. (1954). Continuous inspection schemes. Biometrika, 41, 100–115.MathSciNetCrossRefzbMATHGoogle Scholar
  32. Perron, P., Qu, Z. (2006). Estimating restricted structural change models. Journal of Econometrics, 134, 373–399.Google Scholar
  33. Reeves, J., Chen, J., Wang, X., Lund, R., Lu, Q. (2007). A review and comparison of changepoint detection techniques for climate data. Journal of Applied Meteorology and Climatology, 46, 900–915.Google Scholar
  34. Robbins, M., Lund, R., Gallagher, C., Lu, Q. (2011). Changepoints in the North Atlantic tropical cyclone record. Journal of the American Statistical Association, 106, 89–99.Google Scholar
  35. Rohlfs, R., Harrigan, P., Nielsen, R. (2010). Modeling gene expression evolution with an extended Ornstein-Uhlenbeck process accounting for within-species variation. Scandinavian Journal of Statistics, 37, 200–220.Google Scholar
  36. Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461–464.MathSciNetCrossRefzbMATHGoogle Scholar
  37. Scott, A., Knott, M. (1974). A cluster analysis method for grouping means in the analysis of variance. Biometrics, 30, 507–512.Google Scholar
  38. Sen, A., Srivastava, M. (1975). On tests for detecting change in mean. Annals of Statistics, 3, 98–108.Google Scholar
  39. Shinomoto, S., Sakai, Y., Funahashi, S. (1999). The Ornstein-Uhlenbeck process does not reproduce spiking statistics of neurons in prefrontal cortex. Neural Computation, 11, 935–951.Google Scholar
  40. Shiryaev, A. (1963). On optimum methods in quickest detection problems. Theory of Probability and Its Applications, 8, 26–51.zbMATHGoogle Scholar
  41. Smith, W. (2010). On the simulation and estimation of the mean-reverting Ornstein-Uhlenbeck process. Version 1.01.Google Scholar
  42. Spokoiny, V. (2009). Multiscale local change point detection with applications to value-at-risk. Annals of Statistics, 1405–1436.Google Scholar
  43. Tobing, H., McGilchrist, C. (1992). Recursive residuals for multivariate regression models. Australian Journal of Statistics, 34, 217–232.Google Scholar
  44. Vasicek, O. (1977). An equilibrium characterisation of the term structure. Journal of Financial Economics, 5, 177–188.CrossRefzbMATHGoogle Scholar
  45. Yan, G., Xiao, Z., Eidenbenz, S. (2008). Catching instant messaging worms with change-point detection techniques. In Proceedings of the 1st usenix workshop on large-scale exploits and emergent threats, vol. 6, pp. 1–10.Google Scholar
  46. Zhang, P. (2015). On Stein-rules in generalized mean-reverting processes with change point, Master’s thesis. University of Windsor, Canada.Google Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2017

Authors and Affiliations

  • Fuqi Chen
    • 1
  • Rogemar Mamon
    • 1
    • 2
    Email author
  • Sévérien Nkurunziza
    • 3
  1. 1.Department of Statistical and Actuarial SciencesUniversity of Western OntarioLondonCanada
  2. 2.Division of Physical Sciences and MathematicsUniversity of the Philippines VisayasMiag-ao, IloiloPhilippines
  3. 3.Department of Mathematics and StatisticsUniversity of WindsorWindsorCanada

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